Determinant - Wikipedia

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id="toc-Characterization_of_the_determinant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characterization_of_the_determinant"> <div class="vector-toc-text"> 4.1 Characterization of the determinant </div> </a> <ul id="toc-Characterization_of_the_determinant-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Immediate_consequences" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Immediate_consequences"> <div class="vector-toc-text"> 4.2 Immediate consequences </div> </a> <ul id="toc-Immediate_consequences-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> 4.2.1 Example </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transpose" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transpose"> <div class="vector-toc-text"> 4.3 Transpose </div> </a> <ul id="toc-Transpose-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplicativity_and_matrix_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplicativity_and_matrix_groups"> <div class="vector-toc-text"> 4.4 Multiplicativity and matrix groups </div> </a> <ul id="toc-Multiplicativity_and_matrix_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Laplace_expansion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplace_expansion"> <div class="vector-toc-text"> 4.5 Laplace expansion </div> </a> <ul id="toc-Laplace_expansion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adjugate_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adjugate_matrix"> <div class="vector-toc-text"> 4.6 Adjugate matrix </div> </a> <ul id="toc-Adjugate_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Block_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Block_matrices"> <div class="vector-toc-text"> 4.7 Block matrices </div> </a> <ul id="toc-Block_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sylvester's_determinant_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sylvester's_determinant_theorem"> <div class="vector-toc-text"> 4.8 Sylvester's determinant theorem </div> </a> <ul id="toc-Sylvester's_determinant_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sum"> <div class="vector-toc-text"> 4.9 Sum </div> </a> <ul id="toc-Sum-sublist" class="vector-toc-list"> <li id="toc-Sum_identity_for_2×2_matrices" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sum_identity_for_2×2_matrices"> <div class="vector-toc-text"> 4.9.1 Sum identity for 2×2 matrices </div> </a> <ul id="toc-Sum_identity_for_2×2_matrices-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Properties_of_the_determinant_in_relation_to_other_notions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Properties_of_the_determinant_in_relation_to_other_notions"> <div class="vector-toc-text"> 5 Properties of the determinant in relation to other notions </div> </a> <button aria-controls="toc-Properties_of_the_determinant_in_relation_to_other_notions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Properties of the determinant in relation to other notions subsection </button> <ul id="toc-Properties_of_the_determinant_in_relation_to_other_notions-sublist" class="vector-toc-list"> <li id="toc-Eigenvalues_and_characteristic_polynomial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenvalues_and_characteristic_polynomial"> <div class="vector-toc-text"> 5.1 Eigenvalues and characteristic polynomial </div> </a> <ul id="toc-Eigenvalues_and_characteristic_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trace" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Trace"> <div class="vector-toc-text"> 5.2 Trace </div> </a> <ul id="toc-Trace-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Upper_and_lower_bounds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Upper_and_lower_bounds"> <div class="vector-toc-text"> 5.3 Upper and lower bounds </div> </a> <ul id="toc-Upper_and_lower_bounds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivative"> <div class="vector-toc-text"> 5.4 Derivative </div> </a> <ul id="toc-Derivative-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 6 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 7 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Cramer's_rule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cramer's_rule"> <div class="vector-toc-text"> 7.1 Cramer's rule </div> </a> <ul id="toc-Cramer's_rule-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_independence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_independence"> <div class="vector-toc-text"> 7.2 Linear independence </div> </a> <ul id="toc-Linear_independence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orientation_of_a_basis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orientation_of_a_basis"> <div class="vector-toc-text"> 7.3 Orientation of a basis </div> </a> <ul id="toc-Orientation_of_a_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Volume_and_Jacobian_determinant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volume_and_Jacobian_determinant"> <div class="vector-toc-text"> 7.4 Volume and Jacobian determinant </div> </a> <ul id="toc-Volume_and_Jacobian_determinant-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Abstract_algebraic_aspects" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Abstract_algebraic_aspects"> <div class="vector-toc-text"> 8 Abstract algebraic aspects </div> </a> <button aria-controls="toc-Abstract_algebraic_aspects-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Abstract algebraic aspects subsection </button> <ul id="toc-Abstract_algebraic_aspects-sublist" class="vector-toc-list"> <li id="toc-Determinant_of_an_endomorphism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Determinant_of_an_endomorphism"> <div class="vector-toc-text"> 8.1 Determinant of an endomorphism </div> </a> <ul id="toc-Determinant_of_an_endomorphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Square_matrices_over_commutative_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Square_matrices_over_commutative_rings"> <div class="vector-toc-text"> 8.2 Square matrices over commutative rings </div> </a> <ul id="toc-Square_matrices_over_commutative_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exterior_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exterior_algebra"> <div class="vector-toc-text"> 8.3 Exterior algebra </div> </a> <ul id="toc-Exterior_algebra-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations_and_related_notions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations_and_related_notions"> <div class="vector-toc-text"> 9 Generalizations and related notions </div> </a> <button aria-controls="toc-Generalizations_and_related_notions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations and related notions subsection </button> <ul id="toc-Generalizations_and_related_notions-sublist" class="vector-toc-list"> <li id="toc-Determinants_for_finite-dimensional_algebras" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Determinants_for_finite-dimensional_algebras"> <div class="vector-toc-text"> 9.1 Determinants for finite-dimensional algebras </div> </a> <ul id="toc-Determinants_for_finite-dimensional_algebras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Infinite_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Infinite_matrices"> <div class="vector-toc-text"> 9.2 Infinite matrices </div> </a> <ul id="toc-Infinite_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operators_in_von_Neumann_algebras" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Operators_in_von_Neumann_algebras"> <div class="vector-toc-text"> 9.3 Operators in von Neumann algebras </div> </a> <ul id="toc-Operators_in_von_Neumann_algebras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_notions_for_non-commutative_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Related_notions_for_non-commutative_rings"> <div class="vector-toc-text"> 9.4 Related notions for non-commutative rings </div> </a> <ul id="toc-Related_notions_for_non-commutative_rings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Calculation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculation"> <div class="vector-toc-text"> 10 Calculation </div> </a> <button aria-controls="toc-Calculation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Calculation subsection </button> <ul id="toc-Calculation-sublist" class="vector-toc-list"> <li id="toc-Gaussian_elimination" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gaussian_elimination"> <div class="vector-toc-text"> 10.1 Gaussian elimination </div> </a> <ul id="toc-Gaussian_elimination-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Decomposition_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Decomposition_methods"> <div class="vector-toc-text"> 10.2 Decomposition methods </div> </a> <ul id="toc-Decomposition_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_methods"> <div class="vector-toc-text"> 10.3 Further methods </div> </a> <ul id="toc-Further_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 11 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 12 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 13 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Historical_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_references"> <div class="vector-toc-text"> 13.1 Historical references </div> </a> <ul id="toc-Historical_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Determinant</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 69 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-69" aria-hidden="true" > 69 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AD%D8%AF%D8%AF_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="محدد (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="محدد (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Determinant" title="Determinant – Azerbaijani" lang="az" hreflang="az" data-title="Determinant" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A8%E0%A6%BF%E0%A6%B0%E0%A7%8D%E0%A6%A3%E0%A6%BE%E0%A6%AF%E0%A6%BC%E0%A6%95" title="নির্ণায়ক – Bangla" lang="bn" hreflang="bn" data-title="নির্ণায়ক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/H%C3%A2ng-lia%CC%8Dt-sek" title="Hâng-lia̍t-sek – Minnan" lang="nan" hreflang="nan" data-title="Hâng-lia̍t-sek" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%92%D1%8B%D0%B7%D0%BD%D0%B0%D1%87%D0%BD%D1%96%D0%BA_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Вызначнік (алгебра) – Belarusian" lang="be" hreflang="be" data-title="Вызначнік (алгебра)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D1%82%D0%B5%D1%80%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82%D0%B0" title="Детерминанта – Bulgarian" lang="bg" hreflang="bg" data-title="Детерминанта" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Determinanta" title="Determinanta – Bosnian" lang="bs" hreflang="bs" data-title="Determinanta" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ca.wikipedia.org/wiki/Determinant_(matem%C3%A0tiques)" title="Determinant (matemàtiques) – Catalan" lang="ca" hreflang="ca" data-title="Determinant (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%B0%D0%BB%C4%83%D1%80%D1%82%D0%B0%D0%B2%C3%A7%C4%83" title="Палăртавçă – Chuvash" lang="cv" hreflang="cv" data-title="Палăртавçă" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Determinant" title="Determinant – Czech" lang="cs" hreflang="cs" data-title="Determinant" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Determinant" title="Determinant – Danish" lang="da" hreflang="da" data-title="Determinant" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Determinante" title="Determinante – German" lang="de" hreflang="de" data-title="Determinante" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Determinant" title="Determinant – Estonian" lang="et" hreflang="et" data-title="Determinant" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CF%81%CE%AF%CE%B6%CE%BF%CF%85%CF%83%CE%B1" title="Ορίζουσα – Greek" lang="el" hreflang="el" data-title="Ορίζουσα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Determinante_(matem%C3%A1tica)" title="Determinante (matemática) – Spanish" lang="es" hreflang="es" data-title="Determinante (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Determinanto" title="Determinanto – Esperanto" lang="eo" hreflang="eo" data-title="Determinanto" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Determinante" title="Determinante – Basque" lang="eu" hreflang="eu" data-title="Determinante" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%D8%AA%D8%B1%D9%85%DB%8C%D9%86%D8%A7%D9%86" title="دترمینان – Persian" lang="fa" hreflang="fa" data-title="دترمینان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9terminant_(math%C3%A9matiques)" title="Déterminant (mathématiques) – French" lang="fr" hreflang="fr" data-title="Déterminant (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Deit%C3%A9armanant" title="Deitéarmanant – Irish" lang="ga" hreflang="ga" data-title="Deitéarmanant" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Determinante_(matem%C3%A1ticas)" title="Determinante (matemáticas) – Galician" lang="gl" hreflang="gl" data-title="Determinante (matemáticas)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식 – Korean" lang="ko" hreflang="ko" data-title="행렬식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%88%D6%80%D5%B8%D5%B7%D5%AB%D5%B9_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Որոշիչ (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Որոշիչ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%B0%E0%A4%A3%E0%A4%BF%E0%A4%95" title="सारणिक – Hindi" lang="hi" hreflang="hi" data-title="सारणिक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Determinanta" title="Determinanta – Croatian" lang="hr" hreflang="hr" data-title="Determinanta" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Determinan" title="Determinan – Indonesian" lang="id" hreflang="id" data-title="Determinan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%81kve%C3%B0a" title="Ákveða – Icelandic" lang="is" hreflang="is" data-title="Ákveða" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Determinante_(algebra)" title="Determinante (algebra) – Italian" lang="it" hreflang="it" data-title="Determinante (algebra)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%98%D7%A8%D7%9E%D7%99%D7%A0%D7%A0%D7%98%D7%94" title="דטרמיננטה – Hebrew" lang="he" hreflang="he" data-title="דטרמיננטה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%A8%E0%B2%BF%E0%B2%B0%E0%B3%8D%E0%B2%A7%E0%B2%BE%E0%B2%B0%E0%B2%95" title="ನಿರ್ಧಾರಕ – Kannada" lang="kn" hreflang="kn" data-title="ನಿರ್ಧಾರಕ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%94%E1%83%A2%E1%83%94%E1%83%A0%E1%83%9B%E1%83%98%E1%83%9C%E1%83%90%E1%83%9C%E1%83%A2%E1%83%98" title="დეტერმინანტი – Georgian" lang="ka" hreflang="ka" data-title="დეტერმინანტი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%BD%D1%8B%D2%9B%D1%82%D0%B0%D1%83%D1%8B%D1%88_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Анықтауыш (математика) – Kazakh" lang="kk" hreflang="kk" data-title="Анықтауыш (математика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BD%D1%8B%D0%BA%D1%82%D0%B0%D0%B3%D1%8B%D1%87" title="Аныктагыч – Kyrgyz" lang="ky" hreflang="ky" data-title="Аныктагыч" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Determinans" title="Determinans – Latin" lang="la" hreflang="la" data-title="Determinans" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Determinants" title="Determinants – Latvian" lang="lv" hreflang="lv" data-title="Determinants" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Determinantas" title="Determinantas – Lithuanian" lang="lt" hreflang="lt" data-title="Determinantas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Determinant" title="Determinant – Lombard" lang="lmo" hreflang="lmo" data-title="Determinant" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Determin%C3%A1ns_(matematika)" title="Determináns (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Determináns (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%94%D0%B5%D1%82%D0%B5%D1%80%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82%D0%B0" title="Детерминанта – Macedonian" lang="mk" hreflang="mk" data-title="Детерминанта" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%BE%E0%B4%B0%E0%B4%A3%E0%B4%BF%E0%B4%95%E0%B4%82" title="സാരണികം – Malayalam" lang="ml" hreflang="ml" data-title="സാരണികം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Determinant" title="Determinant – Dutch" lang="nl" hreflang="nl" data-title="Determinant" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式 – Japanese" lang="ja" hreflang="ja" data-title="行列式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Determinant" title="Determinant – Northern Frisian" lang="frr" hreflang="frr" data-title="Determinant" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Determinant" title="Determinant – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Determinant" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Determinant" title="Determinant – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Determinant" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Aniqlovchi_(matematika)" title="Aniqlovchi (matematika) – Uzbek" lang="uz" hreflang="uz" data-title="Aniqlovchi (matematika)" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%88%DB%8C%D9%B9%D8%B1%D9%85%DB%8C%D9%86%D9%86%D9%B9" title="ڈیٹرمیننٹ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ڈیٹرمیننٹ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wyznacznik" title="Wyznacznik – Polish" lang="pl" hreflang="pl" data-title="Wyznacznik" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Determinante" title="Determinante – Portuguese" lang="pt" hreflang="pt" data-title="Determinante" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Determinant_(matematic%C4%83)" title="Determinant (matematică) – Romanian" lang="ro" hreflang="ro" data-title="Determinant (matematică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%BF%D1%80%D0%B5%D0%B4%D0%B5%D0%BB%D0%B8%D1%82%D0%B5%D0%BB%D1%8C" title="Определитель – Russian" lang="ru" hreflang="ru" data-title="Определитель" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Determinanti" title="Determinanti – Albanian" lang="sq" hreflang="sq" data-title="Determinanti" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Determinant" title="Determinant – Simple English" lang="en-simple" hreflang="en-simple" data-title="Determinant" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Determinant_(matematika)" title="Determinant (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Determinant (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Determinanta" title="Determinanta – Slovenian" lang="sl" hreflang="sl" data-title="Determinanta" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AF%DB%8C%D8%AA%DB%8E%D8%B1%D9%85%DB%8C%D9%86%D9%86%D8%AA" title="دیتێرمیننت – Central Kurdish" lang="ckb" hreflang="ckb" data-title="دیتێرمیننت" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B5%D1%82%D0%B5%D1%80%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82%D0%B0" title="Детерминанта – Serbian" lang="sr" hreflang="sr" data-title="Детерминанта" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Determinanta" title="Determinanta – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Determinanta" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Determinantti" title="Determinantti – Finnish" lang="fi" hreflang="fi" data-title="Determinantti" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Determinant" title="Determinant – Swedish" lang="sv" hreflang="sv" data-title="Determinant" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%85%E0%AE%A3%E0%AE%BF%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%B5%E0%AF%88" title="அணிக்கோவை – Tamil" lang="ta" hreflang="ta" data-title="அணிக்கோவை" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%94%E0%B8%B5%E0%B9%80%E0%B8%97%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%A1%E0%B8%B4%E0%B9%81%E0%B8%99%E0%B8%99%E0%B8%95%E0%B9%8C" title="ดีเทอร์มิแนนต์ – Thai" lang="th" hreflang="th" data-title="ดีเทอร์มิแนนต์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Determinant" title="Determinant – Turkish" lang="tr" hreflang="tr" data-title="Determinant" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%B8%D0%B7%D0%BD%D0%B0%D1%87%D0%BD%D0%B8%D0%BA" title="Визначник – Ukrainian" lang="uk" hreflang="uk" data-title="Визначник" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AF%D8%AA%D8%B1%D9%85%DB%8C%D9%86%D8%A7%D9%86" title="دترمینان – Urdu" lang="ur" hreflang="ur" data-title="دترمینان" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BB%8Bnh_th%E1%BB%A9c" title="Định thức – Vietnamese" lang="vi" hreflang="vi" data-title="Định thức" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式 – Wu" lang="wuu" hreflang="wuu" data-title="行列式" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式 – Cantonese" lang="yue" hreflang="yue" data-title="行列式" data-language-autonym="粵語" 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">In mathematics, invariant of square matrices</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about mathematics. For determinants in epidemiology, see <a href="/wiki/Risk_factor" title="Risk factor">Risk factor</a>. For determinants in immunology, see <a href="/wiki/Epitope" title="Epitope">Epitope</a>.</div> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the determinant is a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>-valued <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of the entries of a <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a>. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the <a href="/wiki/Linear_map" title="Linear map">linear map</a> represented, on a given <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>, by the matrix. In particular, the determinant is nonzero <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the matrix is <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> and the corresponding linear map is an <a href="/wiki/Linear_isomorphism" class="mw-redirect" title="Linear isomorphism">isomorphism</a>. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrix</a> is the product of its diagonal entries. The determinant of a 2 × 2 matrix is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1088473ac5aa66c68e3c5075849825b2e53bccc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.849ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc,}"></dd></dl> and the determinant of a 3 × 3 matrix is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mi>e</mi> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mi>g</mi> </mtd> <mtd> <mi>h</mi> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mi>e</mi> <mi>i</mi> <mo>+</mo> <mi>b</mi> <mi>f</mi> <mi>g</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mi>h</mi> <mo>−</mo> <mi>c</mi> <mi>e</mi> <mi>g</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mi>i</mi> <mo>−</mo> <mi>a</mi> <mi>f</mi> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a289e27a018f211758c893b0bdb6c229abb864c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:48.624ex; height:9.509ex;" alt="{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.}"></dd></dl> The determinant of an n × n matrix can be defined in several equivalent ways, the most common being <a href="/wiki/Leibniz_formula_for_determinants" title="Leibniz formula for determinants">Leibniz formula</a>, which expresses the determinant as a sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"> (the <a href="/wiki/Factorial" title="Factorial">factorial</a> of n) signed products of matrix entries. It can be computed by the <a href="/wiki/Laplace_expansion" title="Laplace expansion">Laplace expansion</a>, which expresses the determinant as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of determinants of submatrices, or with <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>, which allows computing a <a href="/wiki/Row_echelon_form" title="Row echelon form">row echelon form</a> with the same determinant, equal to the product of the diagonal entries of the row echelon form. Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the n × n matrices that has the four following properties: <ol><li>The determinant of the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> is 1.</li> <li>The exchange of two rows multiplies the determinant by −1.</li> <li>Multiplying a row by a number multiplies the determinant by this number.</li> <li>Adding a multiple of one row to another row does not change the determinant.</li></ol> The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns. The determinant is invariant under <a href="/wiki/Matrix_similarity" title="Matrix similarity">matrix similarity</a>. This implies that, given a linear <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> of a <a href="/wiki/Finite-dimensional_vector_space" class="mw-redirect" title="Finite-dimensional vector space">finite-dimensional vector space</a>, the determinant of the matrix that represents it on a <a href="/wiki/Basis_(vector_space)" class="mw-redirect" title="Basis (vector space)">basis</a> does not depend on the chosen basis. This allows defining the determinant of a linear endomorphism, which does not depend on the choice of a <a href="/wiki/Coordinate_system" title="Coordinate system">coordinate system</a>. Determinants occur throughout mathematics. For example, a matrix is often used to represent the <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a>, and determinants can be used to solve these equations (<a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a>), although other methods of solution are computationally much more efficient. Determinants are used for defining the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of a square matrix, whose roots are the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>. In <a href="/wiki/Geometry" title="Geometry">geometry</a>, the signed n-dimensional <a href="/wiki/Volume" title="Volume">volume</a> of a n-dimensional <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a> is expressed by a determinant, and the determinant of a linear <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> determines how the <a href="/wiki/Orientability" title="Orientability">orientation</a> and the n-dimensional volume are transformed under the endomorphism. This is used in <a href="/wiki/Calculus" title="Calculus">calculus</a> with <a href="/wiki/Exterior_differential_form" class="mw-redirect" title="Exterior differential form">exterior differential forms</a> and the <a href="/wiki/Jacobian_determinant" class="mw-redirect" title="Jacobian determinant">Jacobian determinant</a>, in particular for <a href="/wiki/Integration_by_substitution#Substitution_for_multiple_variables" title="Integration by substitution">changes of variables</a> in <a href="/wiki/Multiple_integral" title="Multiple integral">multiple integrals</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Two_by_two_matrices">Two by two matrices</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=1" title="Edit section: Two by two matrices">edit</a>]</div> The determinant of a 2 × 2 matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96d54bed2a2ee479cb19f482192d8a0707a0137c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.941ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}a&b\\c&d\end{pmatrix}}}"> is denoted either by "det" or by vertical bars around the matrix, and is defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4116b7a4d1d6ee7c12fbbfa3adc3ca62a8e2d21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.505ex; height:6.176ex;" alt="{\displaystyle \det {\begin{pmatrix}a&b\\c&d\end{pmatrix}}={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.}"></dd></dl> For example, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\begin{pmatrix}3&7\\1&-4\end{pmatrix}}={\begin{vmatrix}3&7\\1&{-4}\end{vmatrix}}=(3\cdot (-4))-(7\cdot 1)=-19.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> </mrow> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>7</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>19.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\begin{pmatrix}3&7\\1&-4\end{pmatrix}}={\begin{vmatrix}3&7\\1&{-4}\end{vmatrix}}=(3\cdot (-4))-(7\cdot 1)=-19.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8241ef945b43ca84326e7cd5de73d2909e4bd5f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.905ex; height:6.176ex;" alt="{\displaystyle \det {\begin{pmatrix}3&7\\1&-4\end{pmatrix}}={\begin{vmatrix}3&7\\1&{-4}\end{vmatrix}}=(3\cdot (-4))-(7\cdot 1)=-19.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="First_properties">First properties</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=2" title="Edit section: First properties">edit</a>]</div> The determinant has several key properties that can be proved by direct evaluation of the definition for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}">-matrices, and that continue to hold for determinants of larger matrices. They are as follows:<a href="#cite_note-1">[1]</a> first, the determinant of the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/520f9c5c868025b50711c4f990ab3401ee76d039" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.82ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&0\\0&1\end{pmatrix}}}"> is 1. Second, the determinant is zero if two rows are the same: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}a&b\\a&b\end{vmatrix}}=ab-ba=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>−</mo> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}a&b\\a&b\end{vmatrix}}=ab-ba=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acdd9610cb800ad88948a454a6ed3b0a68c42f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.896ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}a&b\\a&b\end{vmatrix}}=ab-ba=0.}"></dd></dl> This holds similarly if the two columns are the same. Moreover, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}a&b+b'\\c&d+d'\end{vmatrix}}=a(d+d')-(b+b')c={\begin{vmatrix}a&b\\c&d\end{vmatrix}}+{\begin{vmatrix}a&b'\\c&d'\end{vmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> <mo>+</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo>+</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <msup> <mi>b</mi> <mo>′</mo> </msup> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <msup> <mi>d</mi> <mo>′</mo> </msup> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}a&b+b'\\c&d+d'\end{vmatrix}}=a(d+d')-(b+b')c={\begin{vmatrix}a&b\\c&d\end{vmatrix}}+{\begin{vmatrix}a&b'\\c&d'\end{vmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd9f515d909afb991287394a9afffb309ee6a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.73ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}a&b+b'\\c&d+d'\end{vmatrix}}=a(d+d')-(b+b')c={\begin{vmatrix}a&b\\c&d\end{vmatrix}}+{\begin{vmatrix}a&b'\\c&d'\end{vmatrix}}.}"></dd></dl> Finally, if any column is multiplied by some number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}r\cdot a&b\\r\cdot c&d\end{vmatrix}}=rad-brc=r(ad-bc)=r\cdot {\begin{vmatrix}a&b\\c&d\end{vmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>r</mi> <mo>⋅</mo> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>r</mi> <mo>⋅</mo> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>r</mi> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>r</mi> <mi>c</mi> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}r\cdot a&b\\r\cdot c&d\end{vmatrix}}=rad-brc=r(ad-bc)=r\cdot {\begin{vmatrix}a&b\\c&d\end{vmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c61ebf6b616201373c179bb8f2e33cca4504f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.561ex; height:6.176ex;" alt="{\displaystyle {\begin{vmatrix}r\cdot a&b\\r\cdot c&d\end{vmatrix}}=rad-brc=r(ad-bc)=r\cdot {\begin{vmatrix}a&b\\c&d\end{vmatrix}}.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Geometric_meaning">Geometric meaning</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=3" title="Edit section: Geometric meaning">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Area_parallellogram_as_determinant.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Area_parallellogram_as_determinant.svg/220px-Area_parallellogram_as_determinant.svg.png" decoding="async" width="220" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Area_parallellogram_as_determinant.svg/330px-Area_parallellogram_as_determinant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Area_parallellogram_as_determinant.svg/440px-Area_parallellogram_as_determinant.svg.png 2x" data-file-width="870" data-file-height="1000" /></a><figcaption>The area of the parallelogram is the absolute value of the determinant of the matrix formed by the vectors representing the parallelogram's sides.</figcaption></figure> If the matrix entries are real numbers, the matrix A can be used to represent two <a href="/wiki/Linear_map" title="Linear map">linear maps</a>: one that maps the <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a> vectors to the rows of A, and one that maps them to the columns of A. In either case, the images of the basis vectors form a <a href="/wiki/Parallelogram" title="Parallelogram">parallelogram</a> that represents the image of the <a href="/wiki/Unit_square" title="Unit square">unit square</a> under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at (0, 0), (a, b), (a + c, b + d), and (c, d), as shown in the accompanying diagram. The absolute value of ad − bc is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by A. (The parallelogram formed by the columns of A is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.) The absolute value of the determinant together with the sign becomes the <a href="/wiki/Signed_area" title="Signed area">signed area</a> of the parallelogram. The signed area is the same as the usual <a href="/wiki/Area_(geometry)" class="mw-redirect" title="Area (geometry)">area</a>, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>). To show that ad − bc is the signed area, one may consider a matrix containing two vectors u ≡ (a, b) and v ≡ (c, d) representing the parallelogram's sides. The signed area can be expressed as |u| |v| sin θ for the angle θ between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the <a href="/wiki/Sine" class="mw-redirect" title="Sine">sine</a> this already is the signed area, yet it may be expressed more conveniently using the <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosine</a> of the complementary angle to a perpendicular vector, e.g. u⊥ = (−b, a), so that |u⊥| |v| cos θ′ becomes the signed area in question, which can be determined by the pattern of the <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> to be equal to ad − bc according to the following equations: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Signed area}}=|{\boldsymbol {u}}|\,|{\boldsymbol {v}}|\,\sin \,\theta =\left|{\boldsymbol {u}}^{\perp }\right|\,\left|{\boldsymbol {v}}\right|\,\cos \,\theta '={\begin{pmatrix}-b\\a\end{pmatrix}}\cdot {\begin{pmatrix}c\\d\end{pmatrix}}=ad-bc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Signed area</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>=</mo> <mrow> <mo>|</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mi>cos</mi> <mspace width="thinmathspace" /> <msup> <mi>θ</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mi>d</mi> <mo>−</mo> <mi>b</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Signed area}}=|{\boldsymbol {u}}|\,|{\boldsymbol {v}}|\,\sin \,\theta =\left|{\boldsymbol {u}}^{\perp }\right|\,\left|{\boldsymbol {v}}\right|\,\cos \,\theta '={\begin{pmatrix}-b\\a\end{pmatrix}}\cdot {\begin{pmatrix}c\\d\end{pmatrix}}=ad-bc.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374052fb23aff2aaeeb4f48a6cf0af21a812af5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.524ex; height:6.176ex;" alt="{\displaystyle {\text{Signed area}}=|{\boldsymbol {u}}|\,|{\boldsymbol {v}}|\,\sin \,\theta =\left|{\boldsymbol {u}}^{\perp }\right|\,\left|{\boldsymbol {v}}\right|\,\cos \,\theta '={\begin{pmatrix}-b\\a\end{pmatrix}}\cdot {\begin{pmatrix}c\\d\end{pmatrix}}=ad-bc.}"></dd></dl> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Determinant_parallelepiped.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Determinant_parallelepiped.svg/300px-Determinant_parallelepiped.svg.png" decoding="async" width="300" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Determinant_parallelepiped.svg/450px-Determinant_parallelepiped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Determinant_parallelepiped.svg/600px-Determinant_parallelepiped.svg.png 2x" data-file-width="950" data-file-height="800" /></a><figcaption>The volume of this <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a> is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3.</figcaption></figure> Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by A. When the determinant is equal to one, the linear mapping defined by the matrix is <a href="/wiki/Equiareal_map" title="Equiareal map">equi-areal</a> and orientation-preserving. The object known as the <a href="/wiki/Bivector" title="Bivector">bivector</a> is related to these ideas. In 2D, it can be interpreted as an oriented plane segment formed by imagining two vectors each with origin (0, 0), and coordinates (a, b) and (c, d). The bivector magnitude (denoted by (a, b) ∧ (c, d)) is the signed area, which is also the determinant ad − bc.<a href="#cite_note-2">[2]</a> If an n × n <a href="/wiki/Real_number" title="Real number">real</a> matrix A is written in terms of its column vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left[{\begin{array}{c|c|c|c}\mathbf {a} _{1}&\mathbf {a} _{2}&\cdots &\mathbf {a} _{n}\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em" columnlines="solid solid solid"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left[{\begin{array}{c|c|c|c}\mathbf {a} _{1}&\mathbf {a} _{2}&\cdots &\mathbf {a} _{n}\end{array}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d60337c4cde7b2d5fc3e0365bc8ec5e699ea1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.562ex; height:4.843ex;" alt="{\displaystyle A=\left[{\begin{array}{c|c|c|c}\mathbf {a} _{1}&\mathbf {a} _{2}&\cdots &\mathbf {a} _{n}\end{array}}\right]}">, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}=\mathbf {a} _{1},\quad A{\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}=\mathbf {a} _{2},\quad \ldots ,\quad A{\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}}=\mathbf {a} _{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <mo>…</mo> <mo>,</mo> <mspace width="1em" /> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}=\mathbf {a} _{1},\quad A{\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}=\mathbf {a} _{2},\quad \ldots ,\quad A{\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}}=\mathbf {a} _{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dedbbd65c5f94f4474740dd60c327537f7cddeaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:53.519ex; height:13.843ex;" alt="{\displaystyle A{\begin{pmatrix}1\\0\\\vdots \\0\end{pmatrix}}=\mathbf {a} _{1},\quad A{\begin{pmatrix}0\\1\\\vdots \\0\end{pmatrix}}=\mathbf {a} _{2},\quad \ldots ,\quad A{\begin{pmatrix}0\\0\\\vdots \\1\end{pmatrix}}=\mathbf {a} _{n}.}"></dd></dl> This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> maps the unit <a href="/wiki/Hypercube" title="Hypercube">n-cube</a> to the n-dimensional <a href="/wiki/Parallelepiped#Parallelotope" title="Parallelepiped">parallelotope</a> defined by the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} _{1},\mathbf {a} _{2},\ldots ,\mathbf {a} _{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} _{1},\mathbf {a} _{2},\ldots ,\mathbf {a} _{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6baf696ecee231ce3444b8584f602c14f645bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.084ex; height:2.009ex;" alt="{\displaystyle \mathbf {a} _{1},\mathbf {a} _{2},\ldots ,\mathbf {a} _{n},}"> the region <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\left\{c_{1}\mathbf {a} _{1}+\cdots +c_{n}\mathbf {a} _{n}\mid 0\leq c_{i}\leq 1\ \forall i\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <mn>0</mn> <mo>≤</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mn>1</mn> <mtext> </mtext> <mi mathvariant="normal">∀</mi> <mi>i</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\left\{c_{1}\mathbf {a} _{1}+\cdots +c_{n}\mathbf {a} _{n}\mid 0\leq c_{i}\leq 1\ \forall i\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2b7ef1e3d3f903d706768b53874e8271bde4bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.706ex; height:2.843ex;" alt="{\displaystyle P=\left\{c_{1}\mathbf {a} _{1}+\cdots +c_{n}\mathbf {a} _{n}\mid 0\leq c_{i}\leq 1\ \forall i\right\}.}"> The determinant gives the <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">signed</a> n-dimensional volume of this parallelotope, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\pm {\text{vol}}(P),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>vol</mtext> </mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\pm {\text{vol}}(P),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a71623010b7c6be8e189faef62d7774a3bb1b2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.927ex; height:2.843ex;" alt="{\displaystyle \det(A)=\pm {\text{vol}}(P),}"> and hence describes more generally the n-dimensional volume scaling factor of the <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> produced by A.<a href="#cite_note-3">[3]</a> (The sign shows whether the transformation preserves or reverses <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">orientation</a>.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully n-dimensional, which indicates that the dimension of the image of A is less than n. This <a href="/wiki/Rank%E2%80%93nullity_theorem" title="Rank–nullity theorem">means</a> that A produces a linear transformation which is neither <a href="/wiki/Surjective_function" title="Surjective function">onto</a> nor <a href="/wiki/Injective_function" title="Injective function">one-to-one</a>, and so is not invertible. <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=4" title="Edit section: Definition">edit</a>]</div> Let A be a <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> with n rows and n columns, so that it can be written as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b2d6fc0e5dfa94c40db6b087268a4f733f8ff9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:30.634ex; height:14.843ex;" alt="{\displaystyle A={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{bmatrix}}.}"></dd></dl> The entries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1,1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1,1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/564ac6fe1700e96d1f5acc8123e2faad1d86c8ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.563ex; height:2.343ex;" alt="{\displaystyle a_{1,1}}"> etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>. The determinant of A is denoted by det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{vmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{vmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9cafcfc294a29549ca2d1a6a38e9669407362af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:23.986ex; height:14.843ex;" alt="{\displaystyle {\begin{vmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{vmatrix}}.}"></dd></dl> There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the <a href="/wiki/Leibniz_formula_for_determinants" title="Leibniz formula for determinants">Leibniz formula</a>, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question. <div class="mw-heading mw-heading3"><h3 id="Leibniz_formula">Leibniz formula</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=5" title="Edit section: Leibniz formula">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Leibniz_formula_for_determinants" title="Leibniz formula for determinants">Leibniz formula for determinants</a></div> <div class="mw-heading mw-heading4"><h4 id="3_×_3_matrices">3 × 3 matrices</h4>[<a href="/w/index.php?title=Determinant&action=edit&section=6" title="Edit section: 3 × 3 matrices">edit</a>]</div> The Leibniz formula for the determinant of a 3 × 3 matrix is the following: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mi>e</mi> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mi>g</mi> </mtd> <mtd> <mi>h</mi> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mi>e</mi> <mi>i</mi> <mo>+</mo> <mi>b</mi> <mi>f</mi> <mi>g</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mi>h</mi> <mo>−</mo> <mi>c</mi> <mi>e</mi> <mi>g</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mi>i</mi> <mo>−</mo> <mi>a</mi> <mi>f</mi> <mi>h</mi> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.\ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21563065cb03cf1275650e9d9cfa48314e969463" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:49.592ex; height:9.509ex;" alt="{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=aei+bfg+cdh-ceg-bdi-afh.\ }"></dd></dl> In this expression, each term has one factor from each row, all in different columns, arranged in increasing row order. For example, bdi has b from the first row second column, d from the second row first column, and i from the third row third column. The signs are determined by how many transpositions of factors are necessary to arrange the factors in increasing order of their columns (given that the terms are arranged left-to-right in increasing row order): positive for an even number of transpositions and negative for an odd number. For the example of bdi, the single transposition of bd to db gives dbi, whose three factors are from the first, second and third columns respectively; this is an odd number of transpositions, so the term appears with negative sign. <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sarrus_rule1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Sarrus_rule1.svg/220px-Sarrus_rule1.svg.png" decoding="async" width="220" height="125" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Sarrus_rule1.svg/330px-Sarrus_rule1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Sarrus_rule1.svg/440px-Sarrus_rule1.svg.png 2x" data-file-width="3441" data-file-height="1955" /></a><figcaption><a href="/wiki/Rule_of_Sarrus" title="Rule of Sarrus">Rule of Sarrus</a></figcaption></figure> The <a href="/wiki/Rule_of_Sarrus" title="Rule of Sarrus">rule of Sarrus</a> is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a 3 × 3 matrix does not carry over into higher dimensions. <div class="mw-heading mw-heading4"><h4 id="n_×_n_matrices">n × n matrices</h4>[<a href="/w/index.php?title=Determinant&action=edit&section=7" title="Edit section: n × n matrices">edit</a>]</div> Generalizing the above to higher dimensions, the determinant of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrix is an expression involving <a href="/wiki/Permutation" title="Permutation">permutations</a> and their <a href="/wiki/Signature_(permutation)" class="mw-redirect" title="Signature (permutation)">signatures</a>. A permutation of the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\dots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,\dots ,n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89b43a8431dde909b1835b098104af537dc78125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle \{1,2,\dots ,n\}}"> is a <a href="/wiki/Bijection" title="Bijection">bijective function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> from this set to itself, with values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (1),\sigma (2),\ldots ,\sigma (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (1),\sigma (2),\ldots ,\sigma (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f083feca74458ad8498da95132a268e53443b1fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.349ex; height:2.843ex;" alt="{\displaystyle \sigma (1),\sigma (2),\ldots ,\sigma (n)}"> exhausting the entire set. The set of all such permutations, called the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a>, is commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}">. The signature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(\sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716472e58beb0c37a9bdfbf33851309a7aae45f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(\sigma )}"> of a permutation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d0bb7c02fe831fb1e5e9af84d6d2730b806bad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.617ex; height:2.509ex;" alt="{\displaystyle +1,}"> if the permutation can be obtained with an even number of transpositions (exchanges of two entries); otherwise, it is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1ae9e73ea72a95921a7fbeba221311687f1367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.617ex; height:2.343ex;" alt="{\displaystyle -1.}"> Given a matrix <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> <mspace width="2em" /> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b40704ad9941821164fe63e9249ab604af0a1afe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.62ex; margin-bottom: -0.218ex; width:20.457ex; height:10.843ex;" alt="{\displaystyle A={\begin{bmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{bmatrix}},}"></dd></dl> the Leibniz formula for its determinant is, using <a href="/wiki/Sigma_notation" class="mw-redirect" title="Sigma notation">sigma notation</a> for the sum, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)={\begin{vmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{vmatrix}}=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )a_{1,\sigma (1)}\cdots a_{n,\sigma (n)}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> <mspace width="2em" /> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)={\begin{vmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{vmatrix}}=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )a_{1,\sigma (1)}\cdots a_{n,\sigma (n)}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30368baec488ea7ed0cd3a55bbdb63092fc4251" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.62ex; margin-bottom: -0.218ex; width:53.086ex; height:10.843ex;" alt="{\displaystyle \det(A)={\begin{vmatrix}a_{1,1}\ldots a_{1,n}\\\vdots \qquad \vdots \\a_{n,1}\ldots a_{n,n}\end{vmatrix}}=\sum _{\sigma \in S_{n}}\operatorname {sgn}(\sigma )a_{1,\sigma (1)}\cdots a_{n,\sigma (n)}.}"></dd></dl> Using <a href="/wiki/Pi_notation" class="mw-redirect" title="Pi notation">pi notation</a> for the product, this can be shortened into <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\sum _{\sigma \in S_{n}}\left(\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{i,\sigma (i)}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>∈</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\sum _{\sigma \in S_{n}}\left(\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{i,\sigma (i)}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec63e8d94cbd1c68c2725037f3f5795aa0333a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.485ex; height:7.676ex;" alt="{\displaystyle \det(A)=\sum _{\sigma \in S_{n}}\left(\operatorname {sgn}(\sigma )\prod _{i=1}^{n}a_{i,\sigma (i)}\right)}">.</dd></dl> The <a href="/wiki/Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{i_{1},\ldots ,i_{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{i_{1},\ldots ,i_{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2a87557f939560981e51146e5df02d4954d0c02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.087ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{i_{1},\ldots ,i_{n}}}"> is defined on the n-<a href="/wiki/Tuple" title="Tuple">tuples</a> of integers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,\ldots ,n\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0401c38cf1a2e51b30b38f4b93b5285aa77f8fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.06ex; height:2.843ex;" alt="{\displaystyle \{1,\ldots ,n\}}"> as 0 if two of the integers are equal, and otherwise as the signature of the permutation defined by the n-tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\sum _{i_{1},i_{2},\ldots ,i_{n}}\varepsilon _{i_{1}\cdots i_{n}}a_{1,i_{1}}\!\cdots a_{n,i_{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </munder> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\sum _{i_{1},i_{2},\ldots ,i_{n}}\varepsilon _{i_{1}\cdots i_{n}}a_{1,i_{1}}\!\cdots a_{n,i_{n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2607d3575fab0f19fe6e8418b08ebeb6ae481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:36.404ex; height:5.843ex;" alt="{\displaystyle \det(A)=\sum _{i_{1},i_{2},\ldots ,i_{n}}\varepsilon _{i_{1}\cdots i_{n}}a_{1,i_{1}}\!\cdots a_{n,i_{n}},}"></dd></dl> where the sum is taken over all n-tuples of integers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\ldots ,n\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,\ldots ,n\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b292d5de5f592fa4145ff633eebbbfbb6163a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.707ex; height:2.843ex;" alt="{\displaystyle \{1,\ldots ,n\}.}"> <a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=8" title="Edit section: Properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Characterization_of_the_determinant">Characterization of the determinant</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=9" title="Edit section: Characterization of the determinant">edit</a>]</div> The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}">-matrix A as being composed of its <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> columns, so denoted as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\big (}a_{1},\dots ,a_{n}{\big )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\big (}a_{1},\dots ,a_{n}{\big )},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627aca53fb9f361f6d4fdae1b62bb10e2ff31074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.529ex; height:3.176ex;" alt="{\displaystyle A={\big (}a_{1},\dots ,a_{n}{\big )},}"></dd></dl> where the <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"> (for each i) is composed of the entries of the matrix in the i-th column. <ol><li class="mw-empty-elt"></li><li value="A"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(I\right)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mi>I</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \left(I\right)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01126f85f5125c36d39243643096b6df47e8a796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.859ex; height:2.843ex;" alt="{\displaystyle \det \left(I\right)=1}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is an <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>.</li> <li class="mw-empty-elt"></li><li value="B"> The determinant is <a href="/wiki/Multilinear_map" title="Multilinear map">multilinear</a>: if the jth column of a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is written as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}=r\cdot v+w}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j}=r\cdot v+w}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/284a57001c6971fb7796ece2f8fecc2e6771ab03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.598ex; height:2.676ex;" alt="{\displaystyle a_{j}=r\cdot v+w}"> of two <a href="/wiki/Column_vector" class="mw-redirect" title="Column vector">column vectors</a> v and w and a number r, then the determinant of A is expressible as a similar linear combination: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}|A|&={\big |}a_{1},\dots ,a_{j-1},r\cdot v+w,a_{j+1},\dots ,a_{n}|\\&=r\cdot |a_{1},\dots ,v,\dots a_{n}|+|a_{1},\dots ,w,\dots ,a_{n}|\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>r</mi> <mo>⋅</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>r</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|A|&={\big |}a_{1},\dots ,a_{j-1},r\cdot v+w,a_{j+1},\dots ,a_{n}|\\&=r\cdot |a_{1},\dots ,v,\dots a_{n}|+|a_{1},\dots ,w,\dots ,a_{n}|\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55c8dd7ca89f8057ad32f59f44bffe8e299b71f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.978ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}|A|&={\big |}a_{1},\dots ,a_{j-1},r\cdot v+w,a_{j+1},\dots ,a_{n}|\\&=r\cdot |a_{1},\dots ,v,\dots a_{n}|+|a_{1},\dots ,w,\dots ,a_{n}|\end{aligned}}}"></dd></dl></li> <li class="mw-empty-elt"></li><li value="C">The determinant is <a href="/wiki/Alternating_form" class="mw-redirect" title="Alternating form">alternating</a>: whenever two columns of a matrix are identical, its determinant is 0: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a_{1},\dots ,v,\dots ,v,\dots ,a_{n}|=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>v</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{1},\dots ,v,\dots ,v,\dots ,a_{n}|=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8152b47abddf7f4aa38c01350220a826313dd8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.724ex; height:2.843ex;" alt="{\displaystyle |a_{1},\dots ,v,\dots ,v,\dots ,a_{n}|=0.}"></dd></dl></li></ol> If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}">-matrix A a number that satisfies these three properties.<a href="#cite_note-6">[6]</a> This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula. To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a> vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Immediate_consequences">Immediate consequences</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=10" title="Edit section: Immediate consequences">edit</a>]</div> These rules have several further consequences: <ul><li>The determinant is a <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a>, i.e., <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(cA)=c^{n}\det(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(cA)=c^{n}\det(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6404a766d86e9d78a5c4f82e05de37469a5f8e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.282ex; height:2.843ex;" alt="{\displaystyle \det(cA)=c^{n}\det(A)}"> (for an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">).</li> <li>Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a_{1},\dots ,a_{j},\dots a_{i},\dots ,a_{n}|=-|a_{1},\dots ,a_{i},\dots ,a_{j},\dots ,a_{n}|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{1},\dots ,a_{j},\dots a_{i},\dots ,a_{n}|=-|a_{1},\dots ,a_{i},\dots ,a_{j},\dots ,a_{n}|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba26a4aedda28bb715499a2a38d1c971b85f96e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.979ex; height:3.009ex;" alt="{\displaystyle |a_{1},\dots ,a_{j},\dots a_{i},\dots ,a_{n}|=-|a_{1},\dots ,a_{i},\dots ,a_{j},\dots ,a_{n}|.}"> This formula can be applied iteratively when several columns are swapped. For example <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a_{3},a_{1},a_{2},a_{4}\dots ,a_{n}|=-|a_{1},a_{3},a_{2},a_{4},\dots ,a_{n}|=|a_{1},a_{2},a_{3},a_{4},\dots ,a_{n}|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a_{3},a_{1},a_{2},a_{4}\dots ,a_{n}|=-|a_{1},a_{3},a_{2},a_{4},\dots ,a_{n}|=|a_{1},a_{2},a_{3},a_{4},\dots ,a_{n}|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a52b01c9cec639199b5b71a048ab254fd9cd85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.48ex; height:2.843ex;" alt="{\displaystyle |a_{3},a_{1},a_{2},a_{4}\dots ,a_{n}|=-|a_{1},a_{3},a_{2},a_{4},\dots ,a_{n}|=|a_{1},a_{2},a_{3},a_{4},\dots ,a_{n}|.}"> Yet more generally, any permutation of the columns multiplies the determinant by the <a href="/wiki/Parity_of_a_permutation" title="Parity of a permutation">sign</a> of the permutation.</li> <li>If some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a <a href="/wiki/Linearly_independent" class="mw-redirect" title="Linearly independent">linearly dependent</a> set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0.</li> <li>Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is a <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrix</a>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a565e93211d1ad5c06a571ff8952ab3dfcff3638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:2.843ex;" alt="{\displaystyle a_{ij}=0}">, whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i>j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>></mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i>j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2886c2c870767297c5efcf319aa2bf68a31cb4b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i>j}"> or, alternatively, whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i<j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60ff2d1b23e30fb2979e8c1536da03493f943cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i<j}">, then its determinant equals the product of the diagonal entries: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=a_{11}a_{22}\cdots a_{nn}=\prod _{i=1}^{n}a_{ii}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=a_{11}a_{22}\cdots a_{nn}=\prod _{i=1}^{n}a_{ii}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27643924fe371cdf89a5e12d6646d7aeeccf503e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.723ex; height:6.843ex;" alt="{\displaystyle \det(A)=a_{11}a_{22}\cdots a_{nn}=\prod _{i=1}^{n}a_{ii}.}"> Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> (without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"> which gives a non-zero contribution is the identity permutation.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Example">Example</h4>[<a href="/w/index.php?title=Determinant&action=edit&section=11" title="Edit section: Example">edit</a>]</div> These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> using that method: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}-2&-1&2\\2&1&4\\-3&3&-1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}-2&-1&2\\2&1&4\\-3&3&-1\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f378652bb037bf778741c4f00ceb1c409043e23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.897ex; height:9.176ex;" alt="{\displaystyle A={\begin{bmatrix}-2&-1&2\\2&1&4\\-3&3&-1\end{bmatrix}}.}"></dd></dl> <table class="wikitable"> <caption>Computation of the determinant of matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> </caption> <tbody><tr> <td>Matrix</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\begin{bmatrix}-3&-1&2\\3&1&4\\0&3&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\begin{bmatrix}-3&-1&2\\3&1&4\\0&3&-1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d434149329ef8466edaef8c666500eba2d865057" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.272ex; height:9.176ex;" alt="{\displaystyle B={\begin{bmatrix}-3&-1&2\\3&1&4\\0&3&-1\end{bmatrix}}}"></td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C={\begin{bmatrix}-3&5&2\\3&13&4\\0&0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>13</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\begin{bmatrix}-3&5&2\\3&13&4\\0&0&-1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7bcad51bd8e84cd382e5e8436d0de34d67ecdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.628ex; height:9.176ex;" alt="{\displaystyle C={\begin{bmatrix}-3&5&2\\3&13&4\\0&0&-1\end{bmatrix}}}"> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\begin{bmatrix}5&-3&2\\13&3&4\\0&0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>13</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\begin{bmatrix}5&-3&2\\13&3&4\\0&0&-1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d74b4e56a3fef83fb4a6f4d829bd4608f63da11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.786ex; height:9.176ex;" alt="{\displaystyle D={\begin{bmatrix}5&-3&2\\13&3&4\\0&0&-1\end{bmatrix}}}"> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\begin{bmatrix}18&-3&2\\0&3&4\\0&0&-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>18</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\begin{bmatrix}18&-3&2\\0&3&4\\0&0&-1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca288bd5c3874509b58b2eeb5bf187939f905eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.637ex; height:9.176ex;" alt="{\displaystyle E={\begin{bmatrix}18&-3&2\\0&3&4\\0&0&-1\end{bmatrix}}}"> </td></tr> <tr> <td>Obtained by</td> <td> add the second column to the first </td> <td> add 3 times the third column to the second </td> <td> swap the first two columns </td> <td> add <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {13}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>13</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {13}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db162dc60e438eef414850f6c4679e00f773f1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.969ex; height:5.176ex;" alt="{\displaystyle -{\frac {13}{3}}}"> times the second column to the first </td></tr> <tr> <td>Determinant</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|=|B|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A|=|B|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c27e93c3388b53e40f74667c7471365e3dbc47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle |A|=|B|}"></td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |B|=|C|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |B|=|C|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b7df3776b18b9075a1157ada50f15c319a68eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.216ex; height:2.843ex;" alt="{\displaystyle |B|=|C|}"> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |D|=-|C|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |D|=-|C|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51bea6f0ff58e108f06e4df332e289b3a180df6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.185ex; height:2.843ex;" alt="{\displaystyle |D|=-|C|}"> </td> <td> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E|=|D|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E|=|D|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/851a8c436a7004e4cf32a3f882ba53bc1e4a382a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.386ex; height:2.843ex;" alt="{\displaystyle |E|=|D|}"> </td></tr></tbody></table> Combining these equalities gives <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |A|=-|E|=-(18\cdot 3\cdot (-1))=54.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>18</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>54.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |A|=-|E|=-(18\cdot 3\cdot (-1))=54.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e46a02d1fdb10f0b06cc66f9d58ed68910f599a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.424ex; height:2.843ex;" alt="{\displaystyle |A|=-|E|=-(18\cdot 3\cdot (-1))=54.}"> <div class="mw-heading mw-heading3"><h3 id="Transpose">Transpose</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=12" title="Edit section: Transpose">edit</a>]</div> The determinant of the <a href="/wiki/Transpose" title="Transpose">transpose</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> equals the determinant of A: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(A^{\textsf {T}}\right)=\det(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \left(A^{\textsf {T}}\right)=\det(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e235c54d50da25bb9326c6888ea419c347e8e94b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.721ex; height:3.343ex;" alt="{\displaystyle \det \left(A^{\textsf {T}}\right)=\det(A)}">.</dd></dl> This can be proven by inspecting the Leibniz formula.<a href="#cite_note-7">[7]</a> This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an n × n matrix as being composed of n rows, the determinant is an n-linear function. <div class="mw-heading mw-heading3"><h3 id="Multiplicativity_and_matrix_groups">Multiplicativity and matrix groups</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=13" title="Edit section: Multiplicativity and matrix groups">edit</a>]</div> The determinant is a multiplicative map, i.e., for square matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> of equal size, the determinant of a <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a> equals the product of their determinants: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(AB)=\det(A)\det(B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(AB)=\det(A)\det(B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ae85d6f8ec2ab13701577f53cd24ea669ab506" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.616ex; height:2.843ex;" alt="{\displaystyle \det(AB)=\det(A)\det(B)}"></dd></dl> This key fact can be proven by observing that, for a fixed matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, both sides of the equation are alternating and multilinear as a function depending on the columns of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. Moreover, they both take the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3af4680c0d0d1d09e0f1c4baf24e61954a19b9cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.381ex; height:2.176ex;" alt="{\displaystyle \det B}"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.<a href="#cite_note-8">[8]</a> A matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> with entries in a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> is <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(A^{-1}\right)={\frac {1}{\det(A)}}=[\det(A)]^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \left(A^{-1}\right)={\frac {1}{\det(A)}}=[\det(A)]^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f6798a0a88679c1b82126428cf67aae28244fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.046ex; height:6.009ex;" alt="{\displaystyle \det \left(A^{-1}\right)={\frac {1}{\det(A)}}=[\det(A)]^{-1}}">.</dd></dl> In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">) forms a group known as the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0e7dd529ca93445e5d8071bd0c879d25d43f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.371ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(K)}"> (respectively, a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> called the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SL} _{n}(K)\subset \operatorname {GL} _{n}(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} _{n}(K)\subset \operatorname {GL} _{n}(K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0fbe01d5c4eeacf95d5a6235f0398e53a60532b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.308ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} _{n}(K)\subset \operatorname {GL} _{n}(K)}">. More generally, the word "special" indicates the subgroup of another <a href="/wiki/Matrix_group" class="mw-redirect" title="Matrix group">matrix group</a> of matrices of determinant one. Examples include the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> (which if n is 2 or 3 consists of all <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation matrices</a>), and the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a>. Because the determinant respects multiplication and inverses, it is in fact a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0e7dd529ca93445e5d8071bd0c879d25d43f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.371ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(K)}"> into the multiplicative group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98403ab9e2f27fde4576ca3e622add252f4fd9a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.343ex;" alt="{\displaystyle K^{\times }}"> of nonzero elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. This homomorphism is surjective and its kernel is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SL} _{n}(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} _{n}(K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd1d23901f8cb9734833e006549c29b4df7b4df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.839ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} _{n}(K)}"> (the matrices with determinant one). Hence, by the <a href="/wiki/First_isomorphism_theorem" class="mw-redirect" title="First isomorphism theorem">first isomorphism theorem</a>, this shows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SL} _{n}(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} _{n}(K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd1d23901f8cb9734833e006549c29b4df7b4df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.839ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} _{n}(K)}"> is a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0e7dd529ca93445e5d8071bd0c879d25d43f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.371ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(K)}">, and that the <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(K)/\operatorname {SL} _{n}(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>SL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(K)/\operatorname {SL} _{n}(K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc29b6db2d5d7d5fdc86e1e1dd832d3a0b01817" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.759ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(K)/\operatorname {SL} _{n}(K)}"> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98403ab9e2f27fde4576ca3e622add252f4fd9a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.605ex; height:2.343ex;" alt="{\displaystyle K^{\times }}">. The <a href="/wiki/Cauchy%E2%80%93Binet_formula" title="Cauchy–Binet formula">Cauchy–Binet formula</a> is a generalization of that product formula for rectangular matrices. This formula can also be recast as a multiplicative formula for <a href="/wiki/Compound_matrix" title="Compound matrix">compound matrices</a> whose entries are the determinants of all quadratic submatrices of a given matrix.<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> <div class="mw-heading mw-heading3"><h3 id="Laplace_expansion">Laplace expansion</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=14" title="Edit section: Laplace expansion">edit</a>]</div> <a href="/wiki/Laplace_expansion" title="Laplace expansion">Laplace expansion</a> expresses the determinant of a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> <a href="/wiki/Recursion" title="Recursion">recursively</a> in terms of determinants of smaller matrices, known as its <a href="/wiki/Minor_(matrix)" class="mw-redirect" title="Minor (matrix)">minors</a>. The minor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i,j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0471da86bad57963868302a83a9decf6804d4197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.189ex; height:2.843ex;" alt="{\displaystyle M_{i,j}}"> is defined to be the determinant of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n-1)\times (n-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n-1)\times (n-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db79e223b655326d530528bb7efbc40128589985" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.254ex; height:2.843ex;" alt="{\displaystyle (n-1)\times (n-1)}">-matrix that results from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> by removing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-th row and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}">-th column. The expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)^{i+j}M_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{i+j}M_{i,j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e98b0e073be0bb8e96d378d593bbe918d5ed5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.724ex; height:3.343ex;" alt="{\displaystyle (-1)^{i+j}M_{i,j}}"> is known as a <a href="/wiki/Cofactor_(linear_algebra)" class="mw-redirect" title="Cofactor (linear algebra)">cofactor</a>. For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">, one has the equality <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\sum _{j=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\sum _{j=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de8246be8ba7a7e48ad67871ffef74276fce39ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.771ex; height:7.176ex;" alt="{\displaystyle \det(A)=\sum _{j=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j},}"></dd></dl> which is called the Laplace expansion along the ith row. For example, the Laplace expansion along the first row (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb857369fbc28dab4b5d76bb835e6f196b1d1cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=1}">) gives the following formula: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=a{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c{\begin{vmatrix}d&e\\g&h\end{vmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mi>e</mi> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mi>g</mi> </mtd> <mtd> <mi>h</mi> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>e</mi> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mi>h</mi> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mi>g</mi> </mtd> <mtd> <mi>i</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <mi>g</mi> </mtd> <mtd> <mi>h</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=a{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c{\begin{vmatrix}d&e\\g&h\end{vmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2b66aee07eda072fc4996c89915b0554220226" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:43.321ex; height:9.509ex;" alt="{\displaystyle {\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}=a{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c{\begin{vmatrix}d&e\\g&h\end{vmatrix}}}"></dd></dl> Unwinding the determinants of these <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}">-matrices gives back the Leibniz formula mentioned above. Similarly, the Laplace expansion along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}">-th column is the equality <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\sum _{i=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\sum _{i=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d323ba80a8dd81764153b36e22704c4e28ddd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.771ex; height:6.843ex;" alt="{\displaystyle \det(A)=\sum _{i=1}^{n}(-1)^{i+j}a_{i,j}M_{i,j}.}"></dd></dl> Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the <a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde matrix</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}1&1&1&\cdots &1\\x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\x_{1}^{2}&x_{2}^{2}&x_{3}^{2}&\cdots &x_{n}^{2}\\\vdots &\vdots &\vdots &\ddots &\vdots \\x_{1}^{n-1}&x_{2}^{n-1}&x_{3}^{n-1}&\cdots &x_{n}^{n-1}\end{vmatrix}}=\prod _{1\leq i<j\leq n}\left(x_{j}-x_{i}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo><</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}1&1&1&\cdots &1\\x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\x_{1}^{2}&x_{2}^{2}&x_{3}^{2}&\cdots &x_{n}^{2}\\\vdots &\vdots &\vdots &\ddots &\vdots \\x_{1}^{n-1}&x_{2}^{n-1}&x_{3}^{n-1}&\cdots &x_{n}^{n-1}\end{vmatrix}}=\prod _{1\leq i<j\leq n}\left(x_{j}-x_{i}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb80fd4153db60ff37b612bd5d9f41ea1d82ad8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.505ex; width:53.336ex; height:18.176ex;" alt="{\displaystyle {\begin{vmatrix}1&1&1&\cdots &1\\x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\x_{1}^{2}&x_{2}^{2}&x_{3}^{2}&\cdots &x_{n}^{2}\\\vdots &\vdots &\vdots &\ddots &\vdots \\x_{1}^{n-1}&x_{2}^{n-1}&x_{3}^{n-1}&\cdots &x_{n}^{n-1}\end{vmatrix}}=\prod _{1\leq i<j\leq n}\left(x_{j}-x_{i}\right).}">The n-term Laplace expansion along a row or column can be <a href="/wiki/Laplace_expansion#Laplace_expansion_of_a_determinant_by_complementary_minors" title="Laplace expansion">generalized</a> to write an n x n determinant as a sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tbinom {n}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> <mfrac linethickness="0"> <mi>n</mi> <mi>k</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tbinom {n}{k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206415d3742167e319b2e52c2ca7563b799abad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\displaystyle {\tbinom {n}{k}}}"> <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">terms</a>, each the product of the determinant of a k x k <a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">submatrix</a> and the determinant of the complementary (n−k) x (n−k) submatrix. <div class="mw-heading mw-heading3"><h3 id="Adjugate_matrix">Adjugate matrix</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=15" title="Edit section: Adjugate matrix">edit</a>]</div> The <a href="/wiki/Adjugate_matrix" title="Adjugate matrix">adjugate matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {adj} (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {adj} (A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a03ae409f82f2589e7259fb3d6ff8954fdce65b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.719ex; height:2.843ex;" alt="{\displaystyle \operatorname {adj} (A)}"> is the transpose of the matrix of the cofactors, that is, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\operatorname {adj} (A))_{i,j}=(-1)^{i+j}M_{ji}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msup> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\operatorname {adj} (A))_{i,j}=(-1)^{i+j}M_{ji}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37e07f96f9e1d509334c22faa3d161ab79cc2b41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.475ex; height:3.343ex;" alt="{\displaystyle (\operatorname {adj} (A))_{i,j}=(-1)^{i+j}M_{ji}.}"></dd></dl> For every matrix, one has<a href="#cite_note-11">[11]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\det A)I=A\operatorname {adj} A=(\operatorname {adj} A)\,A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi>I</mi> <mo>=</mo> <mi>A</mi> <mi>adj</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>adj</mi> <mo>⁡</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\det A)I=A\operatorname {adj} A=(\operatorname {adj} A)\,A.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67499f1960e8a408efa9925c27dd0e364f15d7dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.848ex; height:2.843ex;" alt="{\displaystyle (\det A)I=A\operatorname {adj} A=(\operatorname {adj} A)\,A.}"></dd></dl> Thus the adjugate matrix can be used for expressing the inverse of a <a href="/wiki/Nonsingular_matrix" class="mw-redirect" title="Nonsingular matrix">nonsingular matrix</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-1}={\frac {1}{\det A}}\operatorname {adj} A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> </mrow> </mfrac> </mrow> <mi>adj</mi> <mo>⁡</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{-1}={\frac {1}{\det A}}\operatorname {adj} A.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a6a4757c605eac563f8abf39ef00437242b0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.701ex; height:5.343ex;" alt="{\displaystyle A^{-1}={\frac {1}{\det A}}\operatorname {adj} A.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Block_matrices">Block matrices</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=16" title="Edit section: Block matrices">edit</a>]</div> The formula for the determinant of a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}">-matrix above continues to hold, under appropriate further assumptions, for a <a href="/wiki/Block_matrix" title="Block matrix">block matrix</a>, i.e., a matrix composed of four submatrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,B,C,D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</mi> <mo>,</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B,C,D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/684d01c09b12e5a28987c6127567daef29ee3b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.3ex; height:2.509ex;" alt="{\displaystyle A,B,C,D}"> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367523981d714dcd9214703d654bfdedbe58d44a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.921ex; height:1.676ex;" alt="{\displaystyle m\times m}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle n\times m}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}">, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the <a href="/wiki/Schur_complement" title="Schur complement">Schur complement</a>, is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\begin{pmatrix}A&0\\C&D\end{pmatrix}}=\det(A)\det(D)=\det {\begin{pmatrix}A&B\\0&D\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\begin{pmatrix}A&0\\C&D\end{pmatrix}}=\det(A)\det(D)=\det {\begin{pmatrix}A&B\\0&D\end{pmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/572b4824526598a089d188a9bf2aaa14e18aba81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.558ex; height:6.176ex;" alt="{\displaystyle \det {\begin{pmatrix}A&0\\C&D\end{pmatrix}}=\det(A)\det(D)=\det {\begin{pmatrix}A&B\\0&D\end{pmatrix}}.}"></dd></dl> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a>, then it follows with results from the section on multiplicativity that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}&=\det(A)\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}\underbrace {\det {\begin{pmatrix}A^{-1}&-A^{-1}B\\0&I_{n}\end{pmatrix}}} _{=\,\det(A^{-1})\,=\,(\det A)^{-1}}\\&=\det(A)\det {\begin{pmatrix}I_{m}&0\\CA^{-1}&D-CA^{-1}B\end{pmatrix}}\\&=\det(A)\det(D-CA^{-1}B),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> <mspace width="thinmathspace" /> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </munder> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi>D</mi> <mo>−</mo> <mi>C</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mi>C</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}&=\det(A)\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}\underbrace {\det {\begin{pmatrix}A^{-1}&-A^{-1}B\\0&I_{n}\end{pmatrix}}} _{=\,\det(A^{-1})\,=\,(\det A)^{-1}}\\&=\det(A)\det {\begin{pmatrix}I_{m}&0\\CA^{-1}&D-CA^{-1}B\end{pmatrix}}\\&=\det(A)\det(D-CA^{-1}B),\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0019e44c97136c23a4272c061a00510c18a41528" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:60.46ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}&=\det(A)\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}\underbrace {\det {\begin{pmatrix}A^{-1}&-A^{-1}B\\0&I_{n}\end{pmatrix}}} _{=\,\det(A^{-1})\,=\,(\det A)^{-1}}\\&=\det(A)\det {\begin{pmatrix}I_{m}&0\\CA^{-1}&D-CA^{-1}B\end{pmatrix}}\\&=\det(A)\det(D-CA^{-1}B),\end{aligned}}}"></dd></dl> which simplifies to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)(D-CA^{-1}B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mi>C</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)(D-CA^{-1}B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4943463dc13dd2ef835c5ad87e60361c62c170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.962ex; height:3.176ex;" alt="{\displaystyle \det(A)(D-CA^{-1}B)}"> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b4bf91a527dc01af9ef6ace81199becf1308e00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 1\times 1}">-matrix. A similar result holds when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> is invertible, namely <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}&=\det(D)\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}\underbrace {\det {\begin{pmatrix}I_{m}&0\\-D^{-1}C&D^{-1}\end{pmatrix}}} _{=\,\det(D^{-1})\,=\,(\det D)^{-1}}\\&=\det(D)\det {\begin{pmatrix}A-BD^{-1}C&BD^{-1}\\0&I_{n}\end{pmatrix}}\\&=\det(D)\det(A-BD^{-1}C).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>C</mi> </mtd> <mtd> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> <mspace width="thinmathspace" /> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mi>D</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </munder> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> <mo>−</mo> <mi>B</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>C</mi> </mtd> <mtd> <mi>B</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>B</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>C</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}&=\det(D)\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}\underbrace {\det {\begin{pmatrix}I_{m}&0\\-D^{-1}C&D^{-1}\end{pmatrix}}} _{=\,\det(D^{-1})\,=\,(\det D)^{-1}}\\&=\det(D)\det {\begin{pmatrix}A-BD^{-1}C&BD^{-1}\\0&I_{n}\end{pmatrix}}\\&=\det(D)\det(A-BD^{-1}C).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/159b1d8a21ade65b2676eb64474c2936bfc7787e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:61.006ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}&=\det(D)\det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}\underbrace {\det {\begin{pmatrix}I_{m}&0\\-D^{-1}C&D^{-1}\end{pmatrix}}} _{=\,\det(D^{-1})\,=\,(\det D)^{-1}}\\&=\det(D)\det {\begin{pmatrix}A-BD^{-1}C&BD^{-1}\\0&I_{n}\end{pmatrix}}\\&=\det(D)\det(A-BD^{-1}C).\end{aligned}}}"></dd></dl> Both results can be combined to derive <a href="/wiki/Sylvester%27s_determinant_theorem" class="mw-redirect" title="Sylvester's determinant theorem">Sylvester's determinant theorem</a>, which is also stated below. If the blocks are square matrices of the same size further formulas hold. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commute</a> (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle CD=DC}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mi>D</mi> <mo>=</mo> <mi>D</mi> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle CD=DC}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ee02ba56a1a159f7f8651fb80f008351646284" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.48ex; height:2.176ex;" alt="{\displaystyle CD=DC}">), then<a href="#cite_note-12">[12]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(AD-BC).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>C</mi> </mtd> <mtd> <mi>D</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>D</mi> <mo>−</mo> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(AD-BC).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30878ca6402b9074e95deb3f8cf3097b3b655b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.625ex; height:6.176ex;" alt="{\displaystyle \det {\begin{pmatrix}A&B\\C&D\end{pmatrix}}=\det(AD-BC).}"></dd></dl> This formula has been generalized to matrices composed of more than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"> blocks, again under appropriate commutativity conditions among the individual blocks.<a href="#cite_note-13">[13]</a> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/061081a7be1c99ef5318f58ff82b753885cd43ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.766ex; height:2.176ex;" alt="{\displaystyle A=D}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660f51f6f4110c5bc3413bf62687b347fe989596" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.629ex; height:2.176ex;" alt="{\displaystyle B=C}">, the following formula holds (even if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> do not commute)[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det {\begin{pmatrix}A&B\\B&A\end{pmatrix}}=\det(A-B)\det(A+B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>A</mi> </mtd> <mtd> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mi>B</mi> </mtd> <mtd> <mi>A</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det {\begin{pmatrix}A&B\\B&A\end{pmatrix}}=\det(A-B)\det(A+B).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa46964ac3e52a79a15ce9bed8141133272e53ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.545ex; height:6.176ex;" alt="{\displaystyle \det {\begin{pmatrix}A&B\\B&A\end{pmatrix}}=\det(A-B)\det(A+B).}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Sylvester's_determinant_theorem">Sylvester's determinant theorem</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=17" title="Edit section: Sylvester's determinant theorem">edit</a>]</div> <a href="/wiki/Sylvester%27s_determinant_theorem" class="mw-redirect" title="Sylvester's determinant theorem">Sylvester's determinant theorem</a> states that for A, an m × n matrix, and B, an n × m matrix (so that A and B have dimensions allowing them to be multiplied in either order forming a square matrix): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(I_{\mathit {m}}+AB\right)=\det \left(I_{\mathit {n}}+BA\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">m</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>A</mi> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">n</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>B</mi> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \left(I_{\mathit {m}}+AB\right)=\det \left(I_{\mathit {n}}+BA\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0bfc460f0a39f8891b5eca2c29f4b526e5e3ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.458ex; height:2.843ex;" alt="{\displaystyle \det \left(I_{\mathit {m}}+AB\right)=\det \left(I_{\mathit {n}}+BA\right),}"></dd></dl> where Im and In are the m × m and n × n identity matrices, respectively. From this general result several consequences follow. <div><ol style="list-style-type:lower-alpha"><li>For the case of column vector c and row vector r, each with m components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(I_{\mathit {m}}+cr\right)=1+rc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">m</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>c</mi> <mi>r</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \left(I_{\mathit {m}}+cr\right)=1+rc.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f156a3d9d245925e1f3f2f8e97f4958d4c2925d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.725ex; height:2.843ex;" alt="{\displaystyle \det \left(I_{\mathit {m}}+cr\right)=1+rc.}"></dd></dl></li><li>More generally,<a href="#cite_note-14">[14]</a> for any invertible m × m matrix X, <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(X+AB)=\det(X)\det \left(I_{\mathit {n}}+BX^{-1}A\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-mathit" mathvariant="italic">n</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>B</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(X+AB)=\det(X)\det \left(I_{\mathit {n}}+BX^{-1}A\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58ba2a92b1df732877789721b1e1a453693ebe4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.508ex; height:3.343ex;" alt="{\displaystyle \det(X+AB)=\det(X)\det \left(I_{\mathit {n}}+BX^{-1}A\right),}"></dd></li><li>For a column and row vector as above: <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(X+cr)=\det(X)\det \left(1+rX^{-1}c\right)=\det(X)+r\,\operatorname {adj} (X)\,c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <mi>c</mi> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>r</mi> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>r</mi> <mspace width="thinmathspace" /> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(X+cr)=\det(X)\det \left(1+rX^{-1}c\right)=\det(X)+r\,\operatorname {adj} (X)\,c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e440b1153d4601578e84d8e0d290529d4a040414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.331ex; height:3.343ex;" alt="{\displaystyle \det(X+cr)=\det(X)\det \left(1+rX^{-1}c\right)=\det(X)+r\,\operatorname {adj} (X)\,c.}"></dd></li><li>For square matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> of the same size, the matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BA}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8efb97e621ab9b49f8498a49704690bdeb2698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle BA}"> have the same characteristic polynomials (hence the same eigenvalues).</li></ol></div> A generalization is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det \left(Z+AWB\right)=\det \left(Z\right)\det \left(W\right)\det \left(W^{-1}+BZ^{-1}A\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <mi>Z</mi> <mo>+</mo> <mi>A</mi> <mi>W</mi> <mi>B</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mi>W</mi> <mo>)</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>B</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \left(Z+AWB\right)=\det \left(Z\right)\det \left(W\right)\det \left(W^{-1}+BZ^{-1}A\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ced02b0efa603acb4cb142ca795e6a32618e2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:55.705ex; height:3.343ex;" alt="{\displaystyle \det \left(Z+AWB\right)=\det \left(Z\right)\det \left(W\right)\det \left(W^{-1}+BZ^{-1}A\right)}">(see <a href="/wiki/Matrix_determinant_lemma" title="Matrix determinant lemma">Matrix_determinant_lemma</a>), where Z is an m × m invertible matrix and W is an n × n invertible matrix. <div class="mw-heading mw-heading3"><h3 id="Sum">Sum</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=18" title="Edit section: Sum">edit</a>]</div> The determinant of the sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4279cdbd3cb8ec4c3423065d9a7d83a82cfc89e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.348ex; height:2.343ex;" alt="{\displaystyle A+B}"> of two square matrices of the same size is not in general expressible in terms of the determinants of A and of B. However, for <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive semidefinite matrices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> of equal size, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A+B+C)+\det(C)\geq \det(A+C)+\det(B+C){\text{,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>,</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A+B+C)+\det(C)\geq \det(A+C)+\det(B+C){\text{,}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e0801392c512a9b5b73a4409d329cc5b3715fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.023ex; height:2.843ex;" alt="{\displaystyle \det(A+B+C)+\det(C)\geq \det(A+C)+\det(B+C){\text{,}}}"> with the corollary<a href="#cite_note-15">[15]</a><a href="#cite_note-16">[16]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A+B)\geq \det(A)+\det(B){\text{.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A+B)\geq \det(A)+\det(B){\text{.}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ead4bf2a35eb7c463effd2dee2b58046a30387" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.557ex; height:2.843ex;" alt="{\displaystyle \det(A+B)\geq \det(A)+\det(B){\text{.}}}"> <a href="/wiki/Brunn%E2%80%93Minkowski_theorem" title="Brunn–Minkowski theorem">Brunn–Minkowski theorem</a> implies that the nth root of determinant is a <a href="/wiki/Concave_function" title="Concave function">concave function</a>, when restricted to <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a> positive-definite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrices.<a href="#cite_note-17">[17]</a> Therefore, if A and B are Hermitian positive-definite <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrices, one has <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt[{n}]{\det(A+B)}}\geq {\sqrt[{n}]{\det(A)}}+{\sqrt[{n}]{\det(B)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt[{n}]{\det(A+B)}}\geq {\sqrt[{n}]{\det(A)}}+{\sqrt[{n}]{\det(B)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/478b46e4fda4ff92c302349ea35aed28d73f0197" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.528ex; height:4.843ex;" alt="{\displaystyle {\sqrt[{n}]{\det(A+B)}}\geq {\sqrt[{n}]{\det(A)}}+{\sqrt[{n}]{\det(B)}},}"> since the nth root of the determinant is a <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a>. <div class="mw-heading mw-heading4"><h4 id="Sum_identity_for_2×2_matrices">Sum identity for 2×2 matrices</h4>[<a href="/w/index.php?title=Determinant&action=edit&section=19" title="Edit section: Sum identity for 2×2 matrices">edit</a>]</div> For the special case of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"> matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A+B)=\det(A)+\det(B)+{\text{tr}}(A){\text{tr}}(B)-{\text{tr}}(AB).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>tr</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>tr</mtext> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>tr</mtext> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A+B)=\det(A)+\det(B)+{\text{tr}}(A){\text{tr}}(B)-{\text{tr}}(AB).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4991912ec983a0af03cfc077c30619e2e804663d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.129ex; height:2.843ex;" alt="{\displaystyle \det(A+B)=\det(A)+\det(B)+{\text{tr}}(A){\text{tr}}(B)-{\text{tr}}(AB).}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Properties_of_the_determinant_in_relation_to_other_notions">Properties of the determinant in relation to other notions</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=20" title="Edit section: Properties of the determinant in relation to other notions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Eigenvalues_and_characteristic_polynomial">Eigenvalues and characteristic polynomial</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=21" title="Edit section: Eigenvalues and characteristic polynomial">edit</a>]</div> The determinant is closely related to two other central concepts in linear algebra, the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> and the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of a matrix. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> be an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}">-matrix with <a href="/wiki/Complex_number" title="Complex number">complex</a> entries. Then, by the Fundamental Theorem of Algebra, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> must have exactly n <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvalues</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eff16ba4c6886fbff38f2cbb7cafd9ec35c45d36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.605ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{n}}">. (Here it is understood that an eigenvalue with <a href="/wiki/Algebraic_multiplicity" class="mw-redirect" title="Algebraic multiplicity">algebraic multiplicity</a> μ occurs μ times in this list.) Then, it turns out the determinant of A is equal to the product of these eigenvalues, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42699557860330ebb8293b3aef3d8a5b10461ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.027ex; height:6.843ex;" alt="{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}"></dd></dl> The product of all non-zero eigenvalues is referred to as <a href="/wiki/Pseudo-determinant" title="Pseudo-determinant">pseudo-determinant</a>. From this, one immediately sees that the determinant of a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is zero if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> is an eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is invertible if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> is not an eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. The characteristic polynomial is defined as<a href="#cite_note-18">[18]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}(t)=\det(t\cdot I-A).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>⋅</mo> <mi>I</mi> <mo>−</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{A}(t)=\det(t\cdot I-A).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69cd45575f6039fd8997fe05a3cbe3d02387c272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.627ex; height:2.843ex;" alt="{\displaystyle \chi _{A}(t)=\det(t\cdot I-A).}"></dd></dl> Here, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> is the <a href="/wiki/Indeterminate_(variable)" title="Indeterminate (variable)">indeterminate</a> of the polynomial and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is the identity matrix of the same size as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. By means of this polynomial, determinants can be used to find the <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">: they are precisely the <a href="/wiki/Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">roots</a> of this polynomial, i.e., those complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> such that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{A}(\lambda )=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \chi _{A}(\lambda )=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/325549974faf42a21688ef33e8ccbd09d58e9caf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.992ex; height:2.843ex;" alt="{\displaystyle \chi _{A}(\lambda )=0.}"></dd></dl> A <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrix</a> is <a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">positive definite</a> if all its eigenvalues are positive. <a href="/wiki/Sylvester%27s_criterion" title="Sylvester's criterion">Sylvester's criterion</a> asserts that this is equivalent to the determinants of the submatrices <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{k}:={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,k}\\a_{2,1}&a_{2,2}&\cdots &a_{2,k}\\\vdots &\vdots &\ddots &\vdots \\a_{k,1}&a_{k,2}&\cdots &a_{k,k}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{k}:={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,k}\\a_{2,1}&a_{2,2}&\cdots &a_{2,k}\\\vdots &\vdots &\ddots &\vdots \\a_{k,1}&a_{k,2}&\cdots &a_{k,k}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c37842dddf257109f487d9c7bfb903e5c272677" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:31.204ex; height:14.843ex;" alt="{\displaystyle A_{k}:={\begin{bmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,k}\\a_{2,1}&a_{2,2}&\cdots &a_{2,k}\\\vdots &\vdots &\ddots &\vdots \\a_{k,1}&a_{k,2}&\cdots &a_{k,k}\end{bmatrix}}}"></dd></dl> being positive, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.<a href="#cite_note-19">[19]</a> <div class="mw-heading mw-heading3"><h3 id="Trace">Trace</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=22" title="Edit section: Trace">edit</a>]</div> The <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> tr(A) is by definition the sum of the diagonal entries of A and also equals the sum of the eigenvalues. Thus, for complex matrices A, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(\exp(A))=\exp(\operatorname {tr} (A))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(\exp(A))=\exp(\operatorname {tr} (A))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c51da0fef3c61a9ee98f553aa0b8f9591a09c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.972ex; height:2.843ex;" alt="{\displaystyle \det(\exp(A))=\exp(\operatorname {tr} (A))}"></dd></dl> or, for real matrices A, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (A)=\log(\det(\exp(A))).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (A)=\log(\det(\exp(A))).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6796860a3cc4a2cae5469d3a6c659c3e64f3f7a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.039ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (A)=\log(\det(\exp(A))).}"></dd></dl> Here exp(A) denotes the <a href="/wiki/Matrix_exponential" title="Matrix exponential">matrix exponential</a> of A, because every eigenvalue λ of A corresponds to the eigenvalue exp(λ) of exp(A). In particular, given any <a href="/wiki/Matrix_logarithm" class="mw-redirect" title="Matrix logarithm">logarithm</a> of A, that is, any matrix L satisfying <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \exp(L)=A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(L)=A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13ad02578682cc07ace707d9c7fa4799d3d762be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.786ex; height:2.843ex;" alt="{\displaystyle \exp(L)=A}"></dd></dl> the determinant of A is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\exp(\operatorname {tr} (L)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\exp(\operatorname {tr} (L)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/146b9a6059637cd19c349eba15127034414988b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.097ex; height:2.843ex;" alt="{\displaystyle \det(A)=\exp(\operatorname {tr} (L)).}"></dd></dl> For example, for n = 2, n = 3, and n = 4, respectively, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\det(A)&={\frac {1}{2}}\left(\left(\operatorname {tr} (A)\right)^{2}-\operatorname {tr} \left(A^{2}\right)\right),\\\det(A)&={\frac {1}{6}}\left(\left(\operatorname {tr} (A)\right)^{3}-3\operatorname {tr} (A)~\operatorname {tr} \left(A^{2}\right)+2\operatorname {tr} \left(A^{3}\right)\right),\\\det(A)&={\frac {1}{24}}\left(\left(\operatorname {tr} (A)\right)^{4}-6\operatorname {tr} \left(A^{2}\right)\left(\operatorname {tr} (A)\right)^{2}+3\left(\operatorname {tr} \left(A^{2}\right)\right)^{2}+8\operatorname {tr} \left(A^{3}\right)~\operatorname {tr} (A)-6\operatorname {tr} \left(A^{4}\right)\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mrow> <mo>(</mo> <mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>)</mo> </mrow> <mtext> </mtext> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>6</mn> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\det(A)&={\frac {1}{2}}\left(\left(\operatorname {tr} (A)\right)^{2}-\operatorname {tr} \left(A^{2}\right)\right),\\\det(A)&={\frac {1}{6}}\left(\left(\operatorname {tr} (A)\right)^{3}-3\operatorname {tr} (A)~\operatorname {tr} \left(A^{2}\right)+2\operatorname {tr} \left(A^{3}\right)\right),\\\det(A)&={\frac {1}{24}}\left(\left(\operatorname {tr} (A)\right)^{4}-6\operatorname {tr} \left(A^{2}\right)\left(\operatorname {tr} (A)\right)^{2}+3\left(\operatorname {tr} \left(A^{2}\right)\right)^{2}+8\operatorname {tr} \left(A^{3}\right)~\operatorname {tr} (A)-6\operatorname {tr} \left(A^{4}\right)\right).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3435c86907131b735b870199c1b5c81f0a148649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.23ex; margin-bottom: -0.275ex; width:88.121ex; height:16.176ex;" alt="{\displaystyle {\begin{aligned}\det(A)&={\frac {1}{2}}\left(\left(\operatorname {tr} (A)\right)^{2}-\operatorname {tr} \left(A^{2}\right)\right),\\\det(A)&={\frac {1}{6}}\left(\left(\operatorname {tr} (A)\right)^{3}-3\operatorname {tr} (A)~\operatorname {tr} \left(A^{2}\right)+2\operatorname {tr} \left(A^{3}\right)\right),\\\det(A)&={\frac {1}{24}}\left(\left(\operatorname {tr} (A)\right)^{4}-6\operatorname {tr} \left(A^{2}\right)\left(\operatorname {tr} (A)\right)^{2}+3\left(\operatorname {tr} \left(A^{2}\right)\right)^{2}+8\operatorname {tr} \left(A^{3}\right)~\operatorname {tr} (A)-6\operatorname {tr} \left(A^{4}\right)\right).\end{aligned}}}"></dd></dl> cf. <a href="/wiki/Cayley%E2%80%93Hamilton_theorem#Illustration_for_specific_dimensions_and_practical_applications" title="Cayley–Hamilton theorem">Cayley-Hamilton theorem</a>. Such expressions are deducible from combinatorial arguments, <a href="/wiki/Newton%27s_identities#Computing_coefficients" title="Newton's identities">Newton's identities</a>, or the <a href="/wiki/Faddeev%E2%80%93LeVerrier_algorithm" title="Faddeev–LeVerrier algorithm">Faddeev–LeVerrier algorithm</a>. That is, for generic n, detA = (−1)nc0 the signed constant term of the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a>, determined recursively from <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{n}=1;~~~c_{n-m}=-{\frac {1}{m}}\sum _{k=1}^{m}c_{n-m+k}\operatorname {tr} \left(A^{k}\right)~~(1\leq m\leq n)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>;</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>m</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>)</mo> </mrow> <mtext> </mtext> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>≤</mo> <mi>m</mi> <mo>≤</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{n}=1;~~~c_{n-m}=-{\frac {1}{m}}\sum _{k=1}^{m}c_{n-m+k}\operatorname {tr} \left(A^{k}\right)~~(1\leq m\leq n)~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf2468fb3d93f6969039e0cb011c51f108161c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:55.36ex; height:6.843ex;" alt="{\displaystyle c_{n}=1;~~~c_{n-m}=-{\frac {1}{m}}\sum _{k=1}^{m}c_{n-m+k}\operatorname {tr} \left(A^{k}\right)~~(1\leq m\leq n)~.}"></dd></dl> In the general case, this may also be obtained from<a href="#cite_note-20">[20]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\sum _{\begin{array}{c}k_{1},k_{2},\ldots ,k_{n}\geq 0\\k_{1}+2k_{2}+\cdots +nk_{n}=n\end{array}}\prod _{l=1}^{n}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} \left(A^{l}\right)^{k_{l}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>n</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mtd> </mtr> </mtable> </mrow> </munder> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msup> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>!</mo> </mrow> </mfrac> </mrow> <mi>tr</mi> <mo>⁡</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\sum _{\begin{array}{c}k_{1},k_{2},\ldots ,k_{n}\geq 0\\k_{1}+2k_{2}+\cdots +nk_{n}=n\end{array}}\prod _{l=1}^{n}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} \left(A^{l}\right)^{k_{l}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5df716ef3292a1a0504d54058b1a7207d2bc86c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:48.258ex; height:10.176ex;" alt="{\displaystyle \det(A)=\sum _{\begin{array}{c}k_{1},k_{2},\ldots ,k_{n}\geq 0\\k_{1}+2k_{2}+\cdots +nk_{n}=n\end{array}}\prod _{l=1}^{n}{\frac {(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}}\operatorname {tr} \left(A^{l}\right)^{k_{l}},}"></dd></dl> where the sum is taken over the set of all integers kl ≥ 0 satisfying the equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{l=1}^{n}lk_{l}=n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>l</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{l=1}^{n}lk_{l}=n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f2353ac14fbd4081df3cdb4b31084f2939e14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.509ex; height:6.843ex;" alt="{\displaystyle \sum _{l=1}^{n}lk_{l}=n.}"></dd></dl> The formula can be expressed in terms of the complete exponential <a href="/wiki/Bell_polynomial" class="mw-redirect" title="Bell polynomial">Bell polynomial</a> of n arguments sl = −(l – 1)! tr(Al) as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)={\frac {(-1)^{n}}{n!}}B_{n}(s_{1},s_{2},\ldots ,s_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mrow> <mi>n</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)={\frac {(-1)^{n}}{n!}}B_{n}(s_{1},s_{2},\ldots ,s_{n}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91d2e61616ba095eb95e4e9d772cf7d1fe679d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.964ex; height:5.843ex;" alt="{\displaystyle \det(A)={\frac {(-1)^{n}}{n!}}B_{n}(s_{1},s_{2},\ldots ,s_{n}).}"></dd></dl> This formula can also be used to find the determinant of a matrix AIJ with multidimensional indices I = (i1, i2, ..., ir) and J = (j1, j2, ..., jr). The product and trace of such matrices are defined in a natural way as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (AB)_{J}^{I}=\sum _{K}A_{K}^{I}B_{J}^{K},\operatorname {tr} (A)=\sum _{I}A_{I}^{I}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msubsup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munder> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msubsup> <msubsup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msubsup> <mo>,</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </munder> <msubsup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (AB)_{J}^{I}=\sum _{K}A_{K}^{I}B_{J}^{K},\operatorname {tr} (A)=\sum _{I}A_{I}^{I}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f2d3e8806857e31c6fdc4389c437b17eaf8b67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.017ex; height:5.509ex;" alt="{\displaystyle (AB)_{J}^{I}=\sum _{K}A_{K}^{I}B_{J}^{K},\operatorname {tr} (A)=\sum _{I}A_{I}^{I}.}"></dd></dl> An important arbitrary dimension n identity can be obtained from the <a href="/wiki/Mercator_series" title="Mercator series">Mercator series</a> expansion of the logarithm when the expansion converges. If every eigenvalue of A is less than 1 in absolute value, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(I+A)=\sum _{k=0}^{\infty }{\frac {1}{k!}}\left(-\sum _{j=1}^{\infty }{\frac {(-1)^{j}}{j}}\operatorname {tr} \left(A^{j}\right)\right)^{k}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>+</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mi>j</mi> </mfrac> </mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(I+A)=\sum _{k=0}^{\infty }{\frac {1}{k!}}\left(-\sum _{j=1}^{\infty }{\frac {(-1)^{j}}{j}}\operatorname {tr} \left(A^{j}\right)\right)^{k}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31fb89788f5d4224e12ae6de0237c8c6b85f442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:45.582ex; height:8.176ex;" alt="{\displaystyle \det(I+A)=\sum _{k=0}^{\infty }{\frac {1}{k!}}\left(-\sum _{j=1}^{\infty }{\frac {(-1)^{j}}{j}}\operatorname {tr} \left(A^{j}\right)\right)^{k}\,,}"></dd></dl> where I is the identity matrix. More generally, if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k!}}\left(-\sum _{j=1}^{\infty }{\frac {(-1)^{j}s^{j}}{j}}\operatorname {tr} \left(A^{j}\right)\right)^{k}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> </mrow> <mi>j</mi> </mfrac> </mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k!}}\left(-\sum _{j=1}^{\infty }{\frac {(-1)^{j}s^{j}}{j}}\operatorname {tr} \left(A^{j}\right)\right)^{k}\,,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1a6de690bf7a44d1f4d70aa09eabdb147faf74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:33.689ex; height:8.176ex;" alt="{\displaystyle \sum _{k=0}^{\infty }{\frac {1}{k!}}\left(-\sum _{j=1}^{\infty }{\frac {(-1)^{j}s^{j}}{j}}\operatorname {tr} \left(A^{j}\right)\right)^{k}\,,}"></dd></dl> is expanded as a formal <a href="/wiki/Power_series" title="Power series">power series</a> in s then all coefficients of sm for m > n are zero and the remaining polynomial is det(I + sA). <div class="mw-heading mw-heading3"><h3 id="Upper_and_lower_bounds">Upper and lower bounds</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=23" title="Edit section: Upper and lower bounds">edit</a>]</div> For a positive definite matrix A, the trace operator gives the following tight lower and upper bounds on the log determinant <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} \left(I-A^{-1}\right)\leq \log \det(A)\leq \operatorname {tr} (A-I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>I</mi> <mo>−</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <mi>log</mi> <mo>⁡</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \left(I-A^{-1}\right)\leq \log \det(A)\leq \operatorname {tr} (A-I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5012b22044ddb7dd99379ade6d4b6fd6d6d434ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.753ex; height:3.343ex;" alt="{\displaystyle \operatorname {tr} \left(I-A^{-1}\right)\leq \log \det(A)\leq \operatorname {tr} (A-I)}"></dd></dl> with equality if and only if A = I. This relationship can be derived via the formula for the <a href="/wiki/Kullback-Leibler_divergence" class="mw-redirect" title="Kullback-Leibler divergence">Kullback-Leibler divergence</a> between two <a href="/wiki/Multivariate_normal" class="mw-redirect" title="Multivariate normal">multivariate normal</a> distributions. Also, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n}{\operatorname {tr} \left(A^{-1}\right)}}\leq \det(A)^{\frac {1}{n}}\leq {\frac {1}{n}}\operatorname {tr} (A)\leq {\sqrt {{\frac {1}{n}}\operatorname {tr} \left(A^{2}\right)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>≤</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n}{\operatorname {tr} \left(A^{-1}\right)}}\leq \det(A)^{\frac {1}{n}}\leq {\frac {1}{n}}\operatorname {tr} (A)\leq {\sqrt {{\frac {1}{n}}\operatorname {tr} \left(A^{2}\right)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d46d457d03b89698465a4ba6a944cd1cd9abf3f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.803ex; height:6.509ex;" alt="{\displaystyle {\frac {n}{\operatorname {tr} \left(A^{-1}\right)}}\leq \det(A)^{\frac {1}{n}}\leq {\frac {1}{n}}\operatorname {tr} (A)\leq {\sqrt {{\frac {1}{n}}\operatorname {tr} \left(A^{2}\right)}}.}"></dd></dl> These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the <a href="/wiki/Harmonic_mean" title="Harmonic mean">harmonic mean</a> is less than the <a href="/wiki/Geometric_mean" title="Geometric mean">geometric mean</a>, which is less than the <a href="/wiki/Arithmetic_mean" title="Arithmetic mean">arithmetic mean</a>, which is, in turn, less than the <a href="/wiki/Root_mean_square" title="Root mean square">root mean square</a>. <div class="mw-heading mw-heading3"><h3 id="Derivative">Derivative</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=24" title="Edit section: Derivative">edit</a>]</div> The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> function from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} ^{n\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} ^{n\times n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f6e1e2b48aebab7e758e5013224d283c23a8cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.486ex; height:2.343ex;" alt="{\displaystyle \mathbf {R} ^{n\times n}}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }">. In particular, it is everywhere <a href="/wiki/Differentiable" class="mw-redirect" title="Differentiable">differentiable</a>. Its derivative can be expressed using <a href="/wiki/Jacobi%27s_formula" title="Jacobi's formula">Jacobi's formula</a>:<a href="#cite_note-21">[21]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\det(A)}{d\alpha }}=\operatorname {tr} \left(\operatorname {adj} (A){\frac {dA}{d\alpha }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>A</mi> </mrow> <mrow> <mi>d</mi> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\det(A)}{d\alpha }}=\operatorname {tr} \left(\operatorname {adj} (A){\frac {dA}{d\alpha }}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7de92cfccf3afbb2e15ca77637e08a651f14140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.718ex; height:6.343ex;" alt="{\displaystyle {\frac {d\det(A)}{d\alpha }}=\operatorname {tr} \left(\operatorname {adj} (A){\frac {dA}{d\alpha }}\right).}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {adj} (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {adj} (A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a03ae409f82f2589e7259fb3d6ff8954fdce65b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.719ex; height:2.843ex;" alt="{\displaystyle \operatorname {adj} (A)}"> denotes the <a href="/wiki/Adjugate" class="mw-redirect" title="Adjugate">adjugate</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is invertible, we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\det(A)}{d\alpha }}=\det(A)\operatorname {tr} \left(A^{-1}{\frac {dA}{d\alpha }}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>α</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>A</mi> </mrow> <mrow> <mi>d</mi> <mi>α</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\det(A)}{d\alpha }}=\det(A)\operatorname {tr} \left(A^{-1}{\frac {dA}{d\alpha }}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f384d6eaf5819385d15869f66a527bea56980e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.244ex; height:6.343ex;" alt="{\displaystyle {\frac {d\det(A)}{d\alpha }}=\det(A)\operatorname {tr} \left(A^{-1}{\frac {dA}{d\alpha }}\right).}"></dd></dl> Expressed in terms of the entries of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, these are <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \det(A)}{\partial A_{ij}}}=\operatorname {adj} (A)_{ji}=\det(A)\left(A^{-1}\right)_{ji}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \det(A)}{\partial A_{ij}}}=\operatorname {adj} (A)_{ji}=\det(A)\left(A^{-1}\right)_{ji}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb213ad388627083c66cf7fab72692101b2a2643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.828ex; height:6.509ex;" alt="{\displaystyle {\frac {\partial \det(A)}{\partial A_{ij}}}=\operatorname {adj} (A)_{ji}=\det(A)\left(A^{-1}\right)_{ji}.}"></dd></dl> Yet another equivalent formulation is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A+\epsilon X)-\det(A)=\operatorname {tr} (\operatorname {adj} (A)X)\epsilon +O\left(\epsilon ^{2}\right)=\det(A)\operatorname {tr} \left(A^{-1}X\right)\epsilon +O\left(\epsilon ^{2}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>ϵ</mi> <mi>X</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>ϵ</mi> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mi>tr</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>X</mi> </mrow> <mo>)</mo> </mrow> <mi>ϵ</mi> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A+\epsilon X)-\det(A)=\operatorname {tr} (\operatorname {adj} (A)X)\epsilon +O\left(\epsilon ^{2}\right)=\det(A)\operatorname {tr} \left(A^{-1}X\right)\epsilon +O\left(\epsilon ^{2}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e990cbf90ec848510f0e1f3df5afe159f710e89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:78.008ex; height:3.343ex;" alt="{\displaystyle \det(A+\epsilon X)-\det(A)=\operatorname {tr} (\operatorname {adj} (A)X)\epsilon +O\left(\epsilon ^{2}\right)=\det(A)\operatorname {tr} \left(A^{-1}X\right)\epsilon +O\left(\epsilon ^{2}\right)}">,</dd></dl> using <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</a>. The special case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d615ef8214f062617a0c6890d9b69e9ac9b404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.013ex; height:2.176ex;" alt="{\displaystyle A=I}">, the identity matrix, yields <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(I+\epsilon X)=1+\operatorname {tr} (X)\epsilon +O\left(\epsilon ^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>+</mo> <mi>ϵ</mi> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mi>ϵ</mi> <mo>+</mo> <mi>O</mi> <mrow> <mo>(</mo> <msup> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(I+\epsilon X)=1+\operatorname {tr} (X)\epsilon +O\left(\epsilon ^{2}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad9168fa16fad752c8eb75c270ae29917f9828be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.789ex; height:3.343ex;" alt="{\displaystyle \det(I+\epsilon X)=1+\operatorname {tr} (X)\epsilon +O\left(\epsilon ^{2}\right).}"></dd></dl> This identity is used in describing <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a> associated to certain matrix <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>. For example, the special linear group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SL} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a56efb73e5ac1bd83d1d37e83fe94c8a1e51c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.964ex; height:2.509ex;" alt="{\displaystyle \operatorname {SL} _{n}}"> is defined by the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det A=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det A=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20337d5dd614ed9547efbfe48d7e9db3d140e13b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.621ex; height:2.176ex;" alt="{\displaystyle \det A=1}">. The above formula shows that its Lie algebra is the <a href="/wiki/Special_linear_Lie_algebra" title="Special linear Lie algebra">special linear Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c75cbc0e014c41c8cd1e7a40afbadf87eb79de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:2.941ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {sl}}_{n}}"> consisting of those matrices having trace zero. Writing a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc0d4d6106875f8006be1d898512ca5843bad8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}">-matrix as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}a&b&c\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}a&b&c\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c81584d02fde707c6edef738e255a568765539f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.766ex; height:2.843ex;" alt="{\displaystyle A={\begin{bmatrix}a&b&c\end{bmatrix}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"> are column vectors of length 3, then the gradient over one of the three vectors may be written as the <a href="/wiki/Cross_product" title="Cross product">cross product</a> of the other two: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\nabla _{\mathbf {a} }\det(A)&=\mathbf {b} \times \mathbf {c} \\\nabla _{\mathbf {b} }\det(A)&=\mathbf {c} \times \mathbf {a} \\\nabla _{\mathbf {c} }\det(A)&=\mathbf {a} \times \mathbf {b} .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> </msub> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mrow> </msub> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mrow> </msub> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\nabla _{\mathbf {a} }\det(A)&=\mathbf {b} \times \mathbf {c} \\\nabla _{\mathbf {b} }\det(A)&=\mathbf {c} \times \mathbf {a} \\\nabla _{\mathbf {c} }\det(A)&=\mathbf {a} \times \mathbf {b} .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cb0fde5c66f65d1a5924b9f73d08de5459582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:20.509ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\nabla _{\mathbf {a} }\det(A)&=\mathbf {b} \times \mathbf {c} \\\nabla _{\mathbf {b} }\det(A)&=\mathbf {c} \times \mathbf {a} \\\nabla _{\mathbf {c} }\det(A)&=\mathbf {a} \times \mathbf {b} .\end{aligned}}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=25" title="Edit section: History">edit</a>]</div> Historically, determinants were used long before matrices: A determinant was originally defined as a property of a <a href="/wiki/System_of_linear_equations" title="System of linear equations">system of linear equations</a>. The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the Chinese mathematics textbook <a href="/wiki/The_Nine_Chapters_on_the_Mathematical_Art" title="The Nine Chapters on the Mathematical Art">The Nine Chapters on the Mathematical Art</a> (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Cardano</a> in 1545 by a determinant-like entity.<a href="#cite_note-22">[22]</a> Determinants proper originated separately from the work of <a href="/wiki/Seki_Takakazu" title="Seki Takakazu">Seki Takakazu</a> in 1683 in Japan and parallelly of <a href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Leibniz</a> in 1693.<a href="#cite_note-23">[23]</a><a href="#cite_note-Campbell-24">[24]</a><a href="#cite_note-25">[25]</a><a href="#cite_note-26">[26]</a> <a href="#CITEREFCramer1750">Cramer (1750)</a> stated, without proof, Cramer's rule.<a href="#cite_note-27">[27]</a> Both Cramer and also <a href="#CITEREFBézout1779">Bézout (1779)</a> were led to determinants by the question of <a href="/wiki/Plane_curve" title="Plane curve">plane curves</a> passing through a given set of points.<a href="#cite_note-28">[28]</a> <a href="/wiki/Vandermonde" class="mw-redirect" title="Vandermonde">Vandermonde</a> (1771) first recognized determinants as independent functions.<a href="#cite_note-Campbell-24">[24]</a> <a href="#CITEREFLaplace1772">Laplace (1772)</a> gave the general method of expanding a determinant in terms of its complementary <a href="/wiki/Minor_(matrix)" class="mw-redirect" title="Minor (matrix)">minors</a>: Vandermonde had already given a special case.<a href="#cite_note-29">[29]</a> Immediately following, <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Lagrange</a> (1773) treated determinants of the second and third order and applied it to questions of <a href="/wiki/Elimination_theory" title="Elimination theory">elimination theory</a>; he proved many special cases of general identities. <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> (1801) made the next advance. Like Lagrange, he made much use of determinants in the <a href="/wiki/Theory_of_numbers" class="mw-redirect" title="Theory of numbers">theory of numbers</a>. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> of a <a href="/wiki/Algebraic_form" class="mw-redirect" title="Algebraic form">quantic</a>.<a href="#cite_note-30">[30]</a> Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] The next contributor of importance is <a href="/wiki/Jacques_Philippe_Marie_Binet" title="Jacques Philippe Marie Binet">Binet</a> (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of m = n reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, <a href="/wiki/Cauchy" class="mw-redirect" title="Cauchy">Cauchy</a> also presented one on the subject. (See <a href="/wiki/Cauchy%E2%80%93Binet_formula" title="Cauchy–Binet formula">Cauchy–Binet formula</a>.) In this he used the word "determinant" in its present sense,<a href="#cite_note-31">[31]</a><a href="#cite_note-32">[32]</a> summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's.<a href="#cite_note-Campbell-24">[24]</a><a href="#cite_note-33">[33]</a> With him begins the theory in its generality. <a href="#CITEREFJacobi1841">Jacobi (1841)</a> used the functional determinant which Sylvester later called the <a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a>.<a href="#cite_note-34">[34]</a> In his memoirs in <a href="/wiki/Crelle%27s_Journal" title="Crelle's Journal">Crelle's Journal</a> for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called alternants. About the time of Jacobi's last memoirs, <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">Sylvester</a> (1839) and <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a> began their work. <a href="#CITEREFCayley1841">Cayley 1841</a> introduced the modern notation for the determinant using vertical bars.<a href="#cite_note-35">[35]</a><a href="#cite_note-36">[36]</a> The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by <a href="/wiki/Lebesgue" class="mw-redirect" title="Lebesgue">Lebesgue</a>, <a href="/wiki/Otto_Hesse" title="Otto Hesse">Hesse</a>, and Sylvester; <a href="/wiki/Persymmetric" class="mw-redirect" title="Persymmetric">persymmetric</a> determinants by Sylvester and <a href="/wiki/Hermann_Hankel" title="Hermann Hankel">Hankel</a>; <a href="/wiki/Circulant" class="mw-redirect" title="Circulant">circulants</a> by <a href="/wiki/Eug%C3%A8ne_Charles_Catalan" title="Eugène Charles Catalan">Catalan</a>, <a href="/wiki/William_Spottiswoode" title="William Spottiswoode">Spottiswoode</a>, <a href="/wiki/James_Whitbread_Lee_Glaisher" title="James Whitbread Lee Glaisher">Glaisher</a>, and Scott; skew determinants and <a href="/wiki/Pfaffian" title="Pfaffian">Pfaffians</a>, in connection with the theory of <a href="/wiki/Orthogonal_transformation" title="Orthogonal transformation">orthogonal transformation</a>, by Cayley; continuants by Sylvester; <a href="/wiki/Wronskian" title="Wronskian">Wronskians</a> (so called by <a href="/wiki/Thomas_Muir_(mathematician)" title="Thomas Muir (mathematician)">Muir</a>) by <a href="/wiki/Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Christoffel</a> and <a href="/wiki/Ferdinand_Georg_Frobenius" title="Ferdinand Georg Frobenius">Frobenius</a>; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and <a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessians</a> by Sylvester; and symmetric gauche determinants by <a href="/wiki/Trudi" title="Trudi">Trudi</a>. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=26" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Cramer's_rule">Cramer's rule</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=27" title="Edit section: Cramer's rule">edit</a>]</div> Determinants can be used to describe the solutions of a <a href="/wiki/Linear_system_of_equations" class="mw-redirect" title="Linear system of equations">linear system of equations</a>, written in matrix form as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Ax=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax=b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c294fb03a23c833d5b3cc6b3cbe40f25f0005745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.169ex; height:2.176ex;" alt="{\displaystyle Ax=b}">. This equation has a unique solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135eb8f635a86d87cfd1386bc58e3c70a3f8a42a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.782ex; height:2.843ex;" alt="{\displaystyle \det(A)}"> is nonzero. In this case, the solution is given by <a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}={\frac {\det(A_{i})}{\det(A)}}\qquad i=1,2,3,\ldots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}={\frac {\det(A_{i})}{\det(A)}}\qquad i=1,2,3,\ldots ,n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab6a2c36f6405980375f756320d0e0b033a9532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:34.32ex; height:6.509ex;" alt="{\displaystyle x_{i}={\frac {\det(A_{i})}{\det(A)}}\qquad i=1,2,3,\ldots ,n}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"> is the matrix formed by replacing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-th column of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> by the column vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. This follows immediately by column expansion of the determinant, i.e. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A_{i})=\det {\begin{bmatrix}a_{1}&\ldots &b&\ldots &a_{n}\end{bmatrix}}=\sum _{j=1}^{n}x_{j}\det {\begin{bmatrix}a_{1}&\ldots &a_{i-1}&a_{j}&a_{i+1}&\ldots &a_{n}\end{bmatrix}}=x_{i}\det(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo movablelimits="true" form="prefix">det</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A_{i})=\det {\begin{bmatrix}a_{1}&\ldots &b&\ldots &a_{n}\end{bmatrix}}=\sum _{j=1}^{n}x_{j}\det {\begin{bmatrix}a_{1}&\ldots &a_{i-1}&a_{j}&a_{i+1}&\ldots &a_{n}\end{bmatrix}}=x_{i}\det(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2493dcd8dd81d49f8b01dfe52da2f20d60a643ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:98.848ex; height:7.176ex;" alt="{\displaystyle \det(A_{i})=\det {\begin{bmatrix}a_{1}&\ldots &b&\ldots &a_{n}\end{bmatrix}}=\sum _{j=1}^{n}x_{j}\det {\begin{bmatrix}a_{1}&\ldots &a_{i-1}&a_{j}&a_{i+1}&\ldots &a_{n}\end{bmatrix}}=x_{i}\det(A)}"></dd></dl> where the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0096fb78d6843c9fb67a840dc796b61ad93eec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.14ex; height:2.343ex;" alt="{\displaystyle a_{j}}"> are the columns of A. The rule is also implied by the identity <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\,\operatorname {adj} (A)=\operatorname {adj} (A)\,A=\det(A)\,I_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mspace width="thinmathspace" /> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>adj</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>A</mi> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\,\operatorname {adj} (A)=\operatorname {adj} (A)\,A=\det(A)\,I_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07070e564dfa2c0007c5c9c66ab2bff9d9dddd4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.34ex; height:2.843ex;" alt="{\displaystyle A\,\operatorname {adj} (A)=\operatorname {adj} (A)\,A=\det(A)\,I_{n}.}"></dd></dl> Cramer's rule can be implemented in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n^{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b47bb7c0088663b43bea0033e4a9ebc2fbf6340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.066ex; height:3.176ex;" alt="{\displaystyle \operatorname {O} (n^{3})}"> time, which is comparable to more common methods of solving systems of linear equations, such as <a href="/wiki/LU_decomposition" title="LU decomposition">LU</a>, <a href="/wiki/QR_decomposition" title="QR decomposition">QR</a>, or <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a>.<a href="#cite_note-37">[37]</a> <div class="mw-heading mw-heading3"><h3 id="Linear_independence">Linear independence</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=28" title="Edit section: Linear independence">edit</a>]</div> Determinants can be used to characterize <a href="/wiki/Linear_independence" title="Linear independence">linearly dependent</a> vectors: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2d8fe180a2f848cf11e82a535b193cfe718742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.36ex; height:2.176ex;" alt="{\displaystyle \det A}"> is zero if and only if the column vectors (or, equivalently, the row vectors) of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> are linearly dependent.<a href="#cite_note-38">[38]</a> For example, given two linearly independent vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{1},v_{2}\in \mathbf {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1},v_{2}\in \mathbf {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/433e9611c279e1e92960053d957f665adbee4837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.296ex; height:3.009ex;" alt="{\displaystyle v_{1},v_{2}\in \mathbf {R} ^{3}}">, a third vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3237279dde958af2bae661ab58048bda1a9fcdac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{3}}"> lies in the <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> <a href="/wiki/Linear_span" title="Linear span">spanned</a> by the former two vectors exactly if the determinant of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\times 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\times 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddc0d4d6106875f8006be1d898512ca5843bad8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 3\times 3}">-matrix consisting of the three vectors is zero. The same idea is also used in the theory of <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>: given functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(x),\dots ,f_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(x),\dots ,f_{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eac2cff01fc8773051ce03561992e26bc0d754f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.007ex; height:2.843ex;" alt="{\displaystyle f_{1}(x),\dots ,f_{n}(x)}"> (supposed to be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"> times <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable</a>), the <a href="/wiki/Wronskian" title="Wronskian">Wronskian</a> is defined to be <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W(f_{1},\ldots ,f_{n})(x)={\begin{vmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{vmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msubsup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(f_{1},\ldots ,f_{n})(x)={\begin{vmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{vmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bfa62638b23fa3aea4a4f186fd99ae45bb07977" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:60.023ex; height:15.509ex;" alt="{\displaystyle W(f_{1},\ldots ,f_{n})(x)={\begin{vmatrix}f_{1}(x)&f_{2}(x)&\cdots &f_{n}(x)\\f_{1}'(x)&f_{2}'(x)&\cdots &f_{n}'(x)\\\vdots &\vdots &\ddots &\vdots \\f_{1}^{(n-1)}(x)&f_{2}^{(n-1)}(x)&\cdots &f_{n}^{(n-1)}(x)\end{vmatrix}}.}"></dd></dl> It is non-zero (for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">) in a specified interval if and only if the given functions and all their derivatives up to order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"> are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>, this implies the given functions are linearly dependent. See <a href="/wiki/Wronskian#The_Wronskian_and_linear_independence" title="Wronskian">the Wronskian and linear independence</a>. Another such use of the determinant is the <a href="/wiki/Resultant" title="Resultant">resultant</a>, which gives a criterion when two <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> have a common <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a>.<a href="#cite_note-39">[39]</a> <div class="mw-heading mw-heading3"><h3 id="Orientation_of_a_basis">Orientation of a basis</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=29" title="Edit section: Orientation of a basis">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">Orientation (vector space)</a></div> The determinant can be thought of as assigning a number to every <a href="/wiki/Sequence" title="Sequence">sequence</a> of n vectors in Rn, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is a basis for Rn. In that case, the sign of the determinant determines whether the <a href="/wiki/Orientation_(vector_space)" title="Orientation (vector space)">orientation</a> of the basis is consistent with or opposite to the orientation of the <a href="/wiki/Standard_basis" title="Standard basis">standard basis</a>. In the case of an orthogonal basis, the magnitude of the determinant is equal to the product of the lengths of the basis vectors. For instance, an <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a> with entries in Rn represents an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> in <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation. More generally, if the determinant of A is positive, A represents an orientation-preserving <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> (if A is an orthogonal 2 × 2 or 3 × 3 matrix, this is a <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a>), while if it is negative, A switches the orientation of the basis. <div class="mw-heading mw-heading3"><h3 id="Volume_and_Jacobian_determinant">Volume and Jacobian determinant</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=30" title="Edit section: Volume and Jacobian determinant">edit</a>]</div> As pointed out above, the <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> of the determinant of real vectors is equal to the volume of the <a href="/wiki/Parallelepiped" title="Parallelepiped">parallelepiped</a> spanned by those vectors. As a consequence, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/013f555f1d628090325ddff67f7efc027a4cbaf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.273ex; height:2.676ex;" alt="{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{n}}"> is the linear map given by multiplication with a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subset \mathbf {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset \mathbf {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6e3df73059afd5a245b598bfcdfb11d9f2326f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.819ex; height:2.343ex;" alt="{\displaystyle S\subset \mathbf {R} ^{n}}"> is any <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">measurable</a> <a href="/wiki/Subset" title="Subset">subset</a>, then the volume of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a865d36b32ea1d60f15dc3093f5b28093f192b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.587ex; height:2.843ex;" alt="{\displaystyle f(S)}"> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\det(A)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\det(A)|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c97fc5d678940c05b1d84398c75fa444fa49ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.463ex; height:2.843ex;" alt="{\displaystyle |\det(A)|}"> times the volume of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">.<a href="#cite_note-40">[40]</a> More generally, if the linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60aaaf8c1f2d34fc0eb93d2bb237ba49770b7f06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.73ex; height:2.676ex;" alt="{\displaystyle f:\mathbf {R} ^{n}\to \mathbf {R} ^{m}}"> is represented by the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}">-matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, then the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-<a href="/wiki/Dimension" title="Dimension">dimensional</a> volume of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99a865d36b32ea1d60f15dc3093f5b28093f192b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.587ex; height:2.843ex;" alt="{\displaystyle f(S)}"> is given by: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {volume} (f(S))={\sqrt {\det \left(A^{\textsf {T}}A\right)}}\operatorname {volume} (S).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>volume</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo movablelimits="true" form="prefix">det</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>A</mi> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> <mi>volume</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {volume} (f(S))={\sqrt {\det \left(A^{\textsf {T}}A\right)}}\operatorname {volume} (S).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864b2fbe2bf615f12277a5dd3e708ce3edbaf3c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:41.34ex; height:4.843ex;" alt="{\displaystyle \operatorname {volume} (f(S))={\sqrt {\det \left(A^{\textsf {T}}A\right)}}\operatorname {volume} (S).}"></dd></dl> By calculating the volume of the <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a> bounded by four points, they can be used to identify <a href="/wiki/Skew_line" class="mw-redirect" title="Skew line">skew lines</a>. The volume of any tetrahedron, given its <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd0b3d3b09ae6ae430f09c4b317742c56e8acace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.552ex; height:2.509ex;" alt="{\displaystyle a,b,c,d}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{6}}\cdot |\det(a-b,b-c,c-d)|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>,</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo>,</mo> <mi>c</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}\cdot |\det(a-b,b-c,c-d)|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29b4269fb4b1ef59e9e2e39999ffee62f0107f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.441ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}\cdot |\det(a-b,b-c,c-d)|}">, or any other combination of pairs of vertices that form a <a href="/wiki/Spanning_tree" title="Spanning tree">spanning tree</a> over the vertices. <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Jacobian_determinant_and_distortion.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Jacobian_determinant_and_distortion.svg/350px-Jacobian_determinant_and_distortion.svg.png" decoding="async" width="350" height="155" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Jacobian_determinant_and_distortion.svg/525px-Jacobian_determinant_and_distortion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Jacobian_determinant_and_distortion.svg/700px-Jacobian_determinant_and_distortion.svg.png 2x" data-file-width="766" data-file-height="340" /></a><figcaption>A nonlinear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon \mathbf {R} ^{2}\to \mathbf {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon \mathbf {R} ^{2}\to \mathbf {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f528fa02e7789792ac343b356544e6ba0200d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.042ex; height:3.009ex;" alt="{\displaystyle f\colon \mathbf {R} ^{2}\to \mathbf {R} ^{2}}"> sends a small square (left, in red) to a distorted parallelogram (right, in red). The Jacobian at a point gives the best linear approximation of the distorted parallelogram near that point (right, in translucent white), and the Jacobian determinant gives the ratio of the area of the approximating parallelogram to that of the original square.</figcaption></figure> For a general <a href="/wiki/Differentiable_function" title="Differentiable function">differentiable function</a>, much of the above carries over by considering the <a href="/wiki/Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a> of f. For <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c336d25d43bb3a7f40a7a4fe3bf69dec94dbd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.92ex; height:2.676ex;" alt="{\displaystyle f:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n},}"></dd></dl> the Jacobian matrix is the n × n matrix whose entries are given by the <a href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(f)=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{1\leq i,j\leq n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(f)=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{1\leq i,j\leq n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/140cc4cbfdea31575338b57a819dbe6207434ad7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.872ex; height:6.676ex;" alt="{\displaystyle D(f)=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{1\leq i,j\leq n}.}"></dd></dl> Its determinant, the <a href="/wiki/Jacobian_determinant" class="mw-redirect" title="Jacobian determinant">Jacobian determinant</a>, appears in the higher-dimensional version of <a href="/wiki/Integration_by_substitution" title="Integration by substitution">integration by substitution</a>: for suitable functions f and an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> U of Rn (the domain of f), the integral over f(U) of some other function φ : Rn → Rm is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{f(U)}\phi (\mathbf {v} )\,d\mathbf {v} =\int _{U}\phi (f(\mathbf {u} ))\left|\det(\operatorname {D} f)(\mathbf {u} )\right|\,d\mathbf {u} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mrow> <mo>|</mo> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">D</mi> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{f(U)}\phi (\mathbf {v} )\,d\mathbf {v} =\int _{U}\phi (f(\mathbf {u} ))\left|\det(\operatorname {D} f)(\mathbf {u} )\right|\,d\mathbf {u} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78a31a248e5ab8467de8c3865e87c95e762ed3ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:44.592ex; height:6.009ex;" alt="{\displaystyle \int _{f(U)}\phi (\mathbf {v} )\,d\mathbf {v} =\int _{U}\phi (f(\mathbf {u} ))\left|\det(\operatorname {D} f)(\mathbf {u} )\right|\,d\mathbf {u} .}"></dd></dl> The Jacobian also occurs in the <a href="/wiki/Inverse_function_theorem" title="Inverse function theorem">inverse function theorem</a>. When applied to the field of <a href="/wiki/Cartography" title="Cartography">Cartography</a>, the determinant can be used to measure the rate of expansion of a map near the poles.<a href="#cite_note-41">[41]</a> <div class="mw-heading mw-heading2"><h2 id="Abstract_algebraic_aspects">Abstract algebraic aspects </h2>[<a href="/w/index.php?title=Determinant&action=edit&section=31" title="Edit section: Abstract algebraic aspects">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Determinant_of_an_endomorphism">Determinant of an endomorphism</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=32" title="Edit section: Determinant of an endomorphism">edit</a>]</div> The above identities concerning the determinant of products and inverses of matrices imply that <a href="/wiki/Matrix_similarity" title="Matrix similarity">similar matrices</a> have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that A = X−1BX. Indeed, repeatedly applying the above identities yields <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\det(X)^{-1}\det(B)\det(X)=\det(B)\det(X)^{-1}\det(X)=\det(B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\det(X)^{-1}\det(B)\det(X)=\det(B)\det(X)^{-1}\det(X)=\det(B).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/881d426459dcf23a79b2f5440295cab5deb958d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:71.422ex; height:3.176ex;" alt="{\displaystyle \det(A)=\det(X)^{-1}\det(B)\det(X)=\det(B)\det(X)^{-1}\det(X)=\det(B).}"></dd></dl> The determinant is therefore also called a <a href="/wiki/Similarity_invariance" title="Similarity invariance">similarity invariant</a>. The determinant of a <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\to V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fabb548ced097e03f45cf54dbb066d23d010f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.762ex; height:2.176ex;" alt="{\displaystyle T:V\to V}"></dd></dl> for some finite-dimensional <a href="/wiki/Vector_space" title="Vector space">vector space</a> V is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> in V. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T. <div class="mw-heading mw-heading3"><h3 id="Square_matrices_over_commutative_rings">Square matrices over commutative rings</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=33" title="Edit section: Square matrices over commutative rings">edit</a>]</div> The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">, such as the integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776aaf12c2da4b78ca777cb8295c2000bfd51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.634ex; height:2.176ex;" alt="{\displaystyle \mathbf {Z} }">, as opposed to the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(I)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(I)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a167734b1cfb1ec5ad7d1dbd7d1c4471156f97d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.472ex; height:2.843ex;" alt="{\displaystyle \det(I)=1}"> still holds, as do all the properties that result from that characterization.<a href="#cite_note-42">[42]</a> A matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\in \operatorname {Mat} _{n\times n}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <msub> <mi>Mat</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} _{n\times n}(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59b63e08936dc18bfd3d0b56367918c3d5c4c559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.838ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} _{n\times n}(R)}"> is invertible (in the sense that there is an inverse matrix whose entries are in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">) if and only if its determinant is an <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">invertible element</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">.<a href="#cite_note-43">[43]</a> For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\mathbf {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9308b358dfa75f3bf2afc4cd8cbd4f52f224ee33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.496ex; height:2.176ex;" alt="{\displaystyle R=\mathbf {Z} }">, this means that the determinant is +1 or −1. Such a matrix is called <a href="/wiki/Unimodular_matrix" title="Unimodular matrix">unimodular</a>. The determinant being multiplicative, it defines a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(R)\rightarrow R^{\times },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(R)\rightarrow R^{\times },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed1e4e456c303c2a1131a2a7e29de1c9ba52947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.605ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(R)\rightarrow R^{\times },}"></dd></dl> between the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> (the group of invertible <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}">-matrices with entries in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">) and the <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</a> of units in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">. Since it respects the multiplication in both groups, this map is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a>. <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Determinant_as_a_natural_transformation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Determinant_as_a_natural_transformation.svg/300px-Determinant_as_a_natural_transformation.svg.png" decoding="async" width="300" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Determinant_as_a_natural_transformation.svg/450px-Determinant_as_a_natural_transformation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Determinant_as_a_natural_transformation.svg/600px-Determinant_as_a_natural_transformation.svg.png 2x" data-file-width="161" data-file-height="77" /></a><figcaption>The determinant is a natural transformation.</figcaption></figure> Given a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:R\to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9aa7daf2186fbf2fb3b5090e495ad432d4cf42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle f:R\to S}">, there is a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(f):\operatorname {GL} _{n}(R)\to \operatorname {GL} _{n}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(f):\operatorname {GL} _{n}(R)\to \operatorname {GL} _{n}(S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4307d89da56fb6aff7958c16f266e1b5fa017a8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.008ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(f):\operatorname {GL} _{n}(R)\to \operatorname {GL} _{n}(S)}"> given by replacing all entries in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> by their images under <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">. The determinant respects these maps, i.e., the identity <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\det((a_{i,j})))=\det((f(a_{i,j})))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\det((a_{i,j})))=\det((f(a_{i,j})))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9db1db082d7a14d730130af7b68a4dc2f3fc7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.299ex; height:3.009ex;" alt="{\displaystyle f(\det((a_{i,j})))=\det((f(a_{i,j})))}"></dd></dl> holds. In other words, the displayed commutative diagram commutes. For example, the determinant of the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> (the latter determinant being computed using <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>). In the language of <a href="/wiki/Category_theory" title="Category theory">category theory</a>, the determinant is a <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformation</a> between the two functors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/697cc3734cf94b6d8513d6491853969c1b90fbdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.496ex; height:2.509ex;" alt="{\displaystyle \operatorname {GL} _{n}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-)^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-)^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da7b5c0602d0de220122b7993d24c3dc7bc64060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.128ex; height:2.843ex;" alt="{\displaystyle (-)^{\times }}">.<a href="#cite_note-44">[44]</a> Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a>, from the general linear group to the <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det :\operatorname {GL} _{n}\to \mathbb {G} _{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo>:</mo> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det :\operatorname {GL} _{n}\to \mathbb {G} _{m}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e66095f1b7a342d1f6b91a1163fcd3afcf470ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.406ex; height:2.509ex;" alt="{\displaystyle \det :\operatorname {GL} _{n}\to \mathbb {G} _{m}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Exterior_algebra">Exterior algebra</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=34" title="Edit section: Exterior algebra">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Exterior_algebra#Linear_algebra" title="Exterior algebra">Exterior algebra § Linear algebra</a></div> The determinant of a linear transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\to V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\to V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fabb548ced097e03f45cf54dbb066d23d010f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.762ex; height:2.176ex;" alt="{\displaystyle T:V\to V}"> of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-dimensional vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> or, more generally a <a href="/wiki/Free_module" title="Free module">free module</a> of (finite) <a href="/wiki/Rank_of_a_module" class="mw-redirect" title="Rank of a module">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> over a commutative ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> can be formulated in a coordinate-free manner by considering the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-th <a href="/wiki/Exterior_algebra" title="Exterior algebra">exterior power</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge ^{n}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge ^{n}V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40293779e9e9649d07c17d063a6dc281cb94a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.756ex; height:5.176ex;" alt="{\displaystyle \bigwedge ^{n}V}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}">.<a href="#cite_note-45">[45]</a> The map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> induces a linear map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\bigwedge ^{n}T:\bigwedge ^{n}V&\rightarrow \bigwedge ^{n}V\\v_{1}\wedge v_{2}\wedge \dots \wedge v_{n}&\mapsto Tv_{1}\wedge Tv_{2}\wedge \dots \wedge Tv_{n}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>T</mi> <mo>:</mo> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo stretchy="false">→</mo> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>V</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo stretchy="false">↦</mo> <mi>T</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mi>T</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <mi>T</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\bigwedge ^{n}T:\bigwedge ^{n}V&\rightarrow \bigwedge ^{n}V\\v_{1}\wedge v_{2}\wedge \dots \wedge v_{n}&\mapsto Tv_{1}\wedge Tv_{2}\wedge \dots \wedge Tv_{n}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f5a6fa88ba813481212cc41fd091f2ae26eda0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:44.283ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\bigwedge ^{n}T:\bigwedge ^{n}V&\rightarrow \bigwedge ^{n}V\\v_{1}\wedge v_{2}\wedge \dots \wedge v_{n}&\mapsto Tv_{1}\wedge Tv_{2}\wedge \dots \wedge Tv_{n}.\end{aligned}}}"></dd></dl> As <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge ^{n}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge ^{n}V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40293779e9e9649d07c17d063a6dc281cb94a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.756ex; height:5.176ex;" alt="{\displaystyle \bigwedge ^{n}V}"> is one-dimensional, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge ^{n}T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge ^{n}T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f78f8f2400c158f0f59baa4e92f480159f088a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.605ex; height:5.176ex;" alt="{\displaystyle \bigwedge ^{n}T}"> is given by multiplying with some scalar, i.e., an element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">. Some authors such as (<a href="#CITEREFBourbaki1998">Bourbaki 1998</a>) use this fact to define the determinant to be the element in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> satisfying the following identity (for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}\in V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}\in V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/314467aba70c6bd74cacc52d232baacce36f80ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.555ex; height:2.509ex;" alt="{\displaystyle v_{i}\in V}">): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\bigwedge ^{n}T\right)\left(v_{1}\wedge \dots \wedge v_{n}\right)=\det(T)\cdot v_{1}\wedge \dots \wedge v_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>T</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\bigwedge ^{n}T\right)\left(v_{1}\wedge \dots \wedge v_{n}\right)=\det(T)\cdot v_{1}\wedge \dots \wedge v_{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76fef82617eaedbf3718671f3817af280041a559" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.155ex; height:6.343ex;" alt="{\displaystyle \left(\bigwedge ^{n}T\right)\left(v_{1}\wedge \dots \wedge v_{n}\right)=\det(T)\cdot v_{1}\wedge \dots \wedge v_{n}.}"></dd></dl> This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">-tuples of vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ebce0f18894435d56a8fe182e22f135aa8ed07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.982ex; height:2.343ex;" alt="{\displaystyle R^{n}}">. For this reason, the highest non-zero exterior power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge ^{n}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mover> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge ^{n}V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a40293779e9e9649d07c17d063a6dc281cb94a57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.756ex; height:5.176ex;" alt="{\displaystyle \bigwedge ^{n}V}"> (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> and similarly for more involved objects such as <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundles</a> or <a href="/wiki/Chain_complex" title="Chain complex">chain complexes</a> of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigwedge ^{k}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mo>⋀</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </mover> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigwedge ^{k}V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1abd503038dfb7d8e2b67bf9c3302e5afb8b53d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.756ex; height:5.676ex;" alt="{\displaystyle \bigwedge ^{k}V}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k<n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ab7000a6f47e3a09a79dcbe31b89272b0c1f1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.704ex; height:2.176ex;" alt="{\displaystyle k<n}">.<a href="#cite_note-46">[46]</a> <div class="mw-heading mw-heading2"><h2 id="Generalizations_and_related_notions">Generalizations and related notions</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=35" title="Edit section: Generalizations and related notions">edit</a>]</div> Determinants as treated above admit several variants: the <a href="/wiki/Permanent_(mathematics)" title="Permanent (mathematics)">permanent</a> of a matrix is defined as the determinant, except that the factors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn}(\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(\sigma )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716472e58beb0c37a9bdfbf33851309a7aae45f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.51ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn}(\sigma )}"> occurring in Leibniz's rule are omitted. The <a href="/wiki/Immanant_of_a_matrix" class="mw-redirect" title="Immanant of a matrix">immanant</a> generalizes both by introducing a <a href="/wiki/Character_theory" title="Character theory">character</a> of the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}"> in Leibniz's rule. <div class="mw-heading mw-heading3"><h3 id="Determinants_for_finite-dimensional_algebras">Determinants for finite-dimensional algebras</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=36" title="Edit section: Determinants for finite-dimensional algebras">edit</a>]</div> For any <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> that is <a href="/wiki/Dimension" title="Dimension">finite-dimensional</a> as a vector space over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}">, there is a determinant map <a href="#cite_note-47">[47]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det :A\to F.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>F</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det :A\to F.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaa0455160ad4a8223e3e7ce5ce4607124a9b57a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.912ex; height:2.176ex;" alt="{\displaystyle \det :A\to F.}"></dd></dl> This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the <a href="/wiki/Matrix_algebra" class="mw-redirect" title="Matrix algebra">matrix algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\operatorname {Mat} _{n\times n}(F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <msub> <mi>Mat</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\operatorname {Mat} _{n\times n}(F)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a76355c7d8648e3dba0efb6766024ad6ced2b42d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.073ex; height:2.843ex;" alt="{\displaystyle A=\operatorname {Mat} _{n\times n}(F)}">, but also includes several further cases including the determinant of a <a href="/wiki/Quaternion" title="Quaternion">quaternion</a>, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(a+ib+jc+kd)=a^{2}+b^{2}+c^{2}+d^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo>+</mo> <mi>j</mi> <mi>c</mi> <mo>+</mo> <mi>k</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(a+ib+jc+kd)=a^{2}+b^{2}+c^{2}+d^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7b12b7e5ff1dc841cb196da8eb2904c1c403ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.271ex; height:3.176ex;" alt="{\displaystyle \det(a+ib+jc+kd)=a^{2}+b^{2}+c^{2}+d^{2}}">,</dd></dl> the <a href="/wiki/Field_norm" title="Field norm">norm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{L/F}:L\to F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>F</mi> </mrow> </msub> <mo>:</mo> <mi>L</mi> <mo stretchy="false">→</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{L/F}:L\to F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/766ab75bea5896f2271cc0185da5a6e15f0bcbb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.145ex; height:3.009ex;" alt="{\displaystyle N_{L/F}:L\to F}"> of a <a href="/wiki/Field_extension" title="Field extension">field extension</a>, as well as the <a href="/wiki/Pfaffian" title="Pfaffian">Pfaffian</a> of a skew-symmetric matrix and the <a href="/wiki/Reduced_norm" class="mw-redirect" title="Reduced norm">reduced norm</a> of a <a href="/wiki/Central_simple_algebra" title="Central simple algebra">central simple algebra</a>, also arise as special cases of this construction. <div class="mw-heading mw-heading3"><h3 id="Infinite_matrices">Infinite matrices</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=37" title="Edit section: Infinite matrices">edit</a>]</div> For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. <a href="/wiki/Functional_analysis" title="Functional analysis">Functional analysis</a> provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators. The <a href="/wiki/Fredholm_determinant" title="Fredholm determinant">Fredholm determinant</a> defines the determinant for operators known as <a href="/wiki/Trace_class_operator" class="mw-redirect" title="Trace class operator">trace class operators</a> by an appropriate generalization of the formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(I+A)=\exp(\operatorname {tr} (\log(I+A))).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>+</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>+</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(I+A)=\exp(\operatorname {tr} (\log(I+A))).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b67891561bdc833a2142425ea38205dbcb9f481" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.063ex; height:2.843ex;" alt="{\displaystyle \det(I+A)=\exp(\operatorname {tr} (\log(I+A))).}"></dd></dl> Another infinite-dimensional notion of determinant is the <a href="/wiki/Functional_determinant" title="Functional determinant">functional determinant</a>. <div class="mw-heading mw-heading3"><h3 id="Operators_in_von_Neumann_algebras">Operators in von Neumann algebras</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=38" title="Edit section: Operators in von Neumann algebras">edit</a>]</div> For operators in a finite <a href="/wiki/Von_Neumann_algebra#Factors" title="Von Neumann algebra">factor</a>, one may define a positive real-valued determinant called the <a href="/wiki/Fuglede%E2%88%92Kadison_determinant" title="Fuglede−Kadison determinant">Fuglede−Kadison determinant</a> using the canonical trace. In fact, corresponding to every <a href="/wiki/State_(functional_analysis)#tracial_state" title="State (functional analysis)">tracial state</a> on a <a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">von Neumann algebra</a> there is a notion of Fuglede−Kadison determinant. <div class="mw-heading mw-heading3"><h3 id="Related_notions_for_non-commutative_rings">Related notions for non-commutative rings</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=39" title="Edit section: Related notions for non-commutative rings">edit</a>]</div> For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for n ≥ 2,<a href="#cite_note-48">[48]</a> so there is no good definition of the determinant in this setting. For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero bilinear form[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="What exactly is meant by this term must be specified. This statement is valid only if the bilinear form is required to be linear on the same side for both arguments; in contrast, Bourbaki defines a bilinear form B as having the property B(ax,yb) = aB(x,y)b, i.e., left-linear in the left argument and right-linear in the other. (October 2017)">clarify</a>] with a <a href="/wiki/Regular_element_(ring_theory)" class="mw-redirect" title="Regular element (ring theory)">regular element</a> of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably <a href="/wiki/Quasideterminant" title="Quasideterminant">quasideterminants</a> and the <a href="/wiki/Dieudonn%C3%A9_determinant" title="Dieudonné determinant">Dieudonné determinant</a>. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the q-determinant on quantum groups, the <a href="/wiki/Capelli_determinant" class="mw-redirect" title="Capelli determinant">Capelli determinant</a> on Capelli matrices, and the <a href="/wiki/Berezinian" title="Berezinian">Berezinian</a> on <a href="/wiki/Supermatrices" class="mw-redirect" title="Supermatrices">supermatrices</a> (i.e., matrices whose entries are elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92aedfb5c02eff978ab963421ce930f46801657e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{2}}">-<a href="/wiki/Graded_ring" title="Graded ring">graded rings</a>).<a href="#cite_note-49">[49]</a> <a href="/wiki/Manin_matrices" class="mw-redirect" title="Manin matrices">Manin matrices</a> form the class closest to matrices with commutative elements. <div class="mw-heading mw-heading2"><h2 id="Calculation">Calculation</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=40" title="Edit section: Calculation">edit</a>]</div> Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in <a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">numerical linear algebra</a>, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques.<a href="#cite_note-50">[50]</a> <a href="/wiki/Computational_geometry" title="Computational geometry">Computational geometry</a>, however, does frequently use calculations related to determinants.<a href="#cite_note-51">[51]</a> While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> <a href="/wiki/Factorial" title="Factorial">factorial</a>) products for an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}">-matrix. Thus, the number of required operations grows very quickly: it is <a href="/wiki/Big_O_notation" title="Big O notation">of order</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}">. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants. <div class="mw-heading mw-heading3"><h3 id="Gaussian_elimination">Gaussian elimination</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=41" title="Edit section: Gaussian elimination">edit</a>]</div> <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> consists of left multiplying a matrix by <a href="/wiki/Elementary_matrices" class="mw-redirect" title="Elementary matrices">elementary matrices</a> for getting a matrix in a <a href="/wiki/Row_echelon_form" title="Row echelon form">row echelon form</a>. One can restrict the computation to elementary matrices of determinant 1. In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is a <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrix</a>, its determinant is the product of the entries of its diagonal. So, the determinant can be computed for almost free from the result of a Gaussian elimination. <div class="mw-heading mw-heading3"><h3 id="Decomposition_methods">Decomposition methods</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=42" title="Edit section: Decomposition methods">edit</a>]</div> Some methods compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135eb8f635a86d87cfd1386bc58e3c70a3f8a42a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.782ex; height:2.843ex;" alt="{\displaystyle \det(A)}"> by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the <a href="/wiki/LU_decomposition" title="LU decomposition">LU decomposition</a>, the <a href="/wiki/QR_decomposition" title="QR decomposition">QR decomposition</a> or the <a href="/wiki/Cholesky_decomposition" title="Cholesky decomposition">Cholesky decomposition</a> (for <a href="/wiki/Positive_definite_matrix" class="mw-redirect" title="Positive definite matrix">positive definite matrices</a>). These methods are of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n^{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b47bb7c0088663b43bea0033e4a9ebc2fbf6340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.066ex; height:3.176ex;" alt="{\displaystyle \operatorname {O} (n^{3})}">, which is a significant improvement over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>!</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n!)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dae778eddcf984a698656ac26d2495659cf4690a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.659ex; height:2.843ex;" alt="{\displaystyle \operatorname {O} (n!)}">.<a href="#cite_note-52">[52]</a> For example, LU decomposition expresses <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> as a product <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=PLU.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>P</mi> <mi>L</mi> <mi>U</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=PLU.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24103d6156fc58fb2d2427eb14ae70c5804726f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.599ex; height:2.176ex;" alt="{\displaystyle A=PLU.}"></dd></dl> of a <a href="/wiki/Permutation_matrix" title="Permutation matrix">permutation matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> (which has exactly a single <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> in each column, and otherwise zeros), a lower triangular matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> and an upper triangular matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">. The determinants of the two triangular matrices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> is just the sign <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }"> of the corresponding permutation (which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d04cf05c67d41d9f39dabf6a90722ce860a76958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle +1}"> for an even number of permutations and is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -1}"> for an odd number of permutations). Once such a LU decomposition is known for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, its determinant is readily computed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\varepsilon \det(L)\cdot \det(U).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ε</mi> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\varepsilon \det(L)\cdot \det(U).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4191c1c397a75268d051976643998eb33cf0cdcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.12ex; height:2.843ex;" alt="{\displaystyle \det(A)=\varepsilon \det(L)\cdot \det(U).}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Further_methods">Further methods</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=43" title="Edit section: Further methods">edit</a>]</div> The order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n^{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b47bb7c0088663b43bea0033e4a9ebc2fbf6340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.066ex; height:3.176ex;" alt="{\displaystyle \operatorname {O} (n^{3})}"> reached by decomposition methods has been improved by different methods. If two matrices of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> can be multiplied in time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c856803491e153263b91aacc660e26ef129d467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.646ex; height:2.843ex;" alt="{\displaystyle M(n)}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(n)\geq n^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M(n)\geq n^{a}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa04ef1e158e749930e873807bac7e1142de225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.241ex; height:2.843ex;" alt="{\displaystyle M(n)\geq n^{a}}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbdadbae926a492fc3d5f982b0a987ce7e2a416" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>2}">, then there is an algorithm computing the determinant in time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(M(n))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(M(n))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56f3c09fa14dce82d17055cf1b98b7679d6acd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.229ex; height:2.843ex;" alt="{\displaystyle O(M(n))}">.<a href="#cite_note-53">[53]</a> This means, for example, that an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n^{2.376})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2.376</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n^{2.376})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2968ed89e318e8a2755a245b4f5574e15cd8635a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.99ex; height:3.176ex;" alt="{\displaystyle \operatorname {O} (n^{2.376})}"> algorithm for computing the determinant exists based on the <a href="/wiki/Coppersmith%E2%80%93Winograd_algorithm" class="mw-redirect" title="Coppersmith–Winograd algorithm">Coppersmith–Winograd algorithm</a>. This exponent has been further lowered, as of 2016, to 2.373.<a href="#cite_note-54">[54]</a> In addition to the complexity of the algorithm, further criteria can be used to compare algorithms. Especially for applications concerning matrices over rings, algorithms that compute the determinant without any divisions exist. (By contrast, Gauss elimination requires divisions.) One such algorithm, having complexity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n^{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n^{4})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/355bc603ee802ccf4c4f4de551dd76bc2253e2e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.066ex; height:3.176ex;" alt="{\displaystyle \operatorname {O} (n^{4})}"> is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-called <a href="/w/index.php?title=Closed_ordered_walk&action=edit&redlink=1" class="new" title="Closed ordered walk (page does not exist)">closed ordered walks</a>, in which several items can be repeated. The resulting sum has more terms than in the Leibniz rule, but in the process several of these products can be reused, making it more efficient than naively computing with the Leibniz rule.<a href="#cite_note-55">[55]</a> Algorithms can also be assessed according to their <a href="/wiki/Bit_complexity" class="mw-redirect" title="Bit complexity">bit complexity</a>, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> (or LU decomposition) method is of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {O} (n^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">O</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {O} (n^{3})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b47bb7c0088663b43bea0033e4a9ebc2fbf6340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.066ex; height:3.176ex;" alt="{\displaystyle \operatorname {O} (n^{3})}">, but the bit length of intermediate values can become exponentially long.<a href="#cite_note-56">[56]</a> By comparison, the <a href="/wiki/Bareiss_Algorithm" class="mw-redirect" title="Bareiss Algorithm">Bareiss Algorithm</a>, is an exact-division method (so it does use division, but only in cases where these divisions can be performed without remainder) is of the same order, but the bit complexity is roughly the bit size of the original entries in the matrix times <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.<a href="#cite_note-57">[57]</a> If the determinant of A and the inverse of A have already been computed, the <a href="/wiki/Matrix_determinant_lemma" title="Matrix determinant lemma">matrix determinant lemma</a> allows rapid calculation of the determinant of A + uvT, where u and v are column vectors. Charles Dodgson (i.e. <a href="/wiki/Lewis_Carroll" title="Lewis Carroll">Lewis Carroll</a> of <a href="/wiki/Alice%27s_Adventures_in_Wonderland" title="Alice's Adventures in Wonderland">Alice's Adventures in Wonderland</a> fame) invented a method for computing determinants called <a href="/wiki/Dodgson_condensation" title="Dodgson condensation">Dodgson condensation</a>. Unfortunately this interesting method does not always work in its original form.<a href="#cite_note-58">[58]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=44" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239009302">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Cauchy_determinant" class="mw-redirect" title="Cauchy determinant">Cauchy determinant</a></li> <li><a href="/wiki/Cayley%E2%80%93Menger_determinant" title="Cayley–Menger determinant">Cayley–Menger determinant</a></li> <li><a href="/wiki/Dieudonn%C3%A9_determinant" title="Dieudonné determinant">Dieudonné determinant</a></li> <li><a href="/wiki/Slater_determinant" title="Slater determinant">Slater determinant</a></li> <li><a href="/wiki/Determinantal_conjecture" title="Determinantal conjecture">Determinantal conjecture</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=45" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFLang1985">Lang 1985</a>, §VII.1 </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWildberger2010" class="citation audio-visual cs1">Wildberger, Norman J. (2010). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=6XghF70fqkY">Episode 4</a> (video lecture). WildLinAlg. Sydney, Australia: <a href="/wiki/University_of_New_South_Wales" title="University of New South Wales">University of New South Wales</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211211/6XghF70fqkY">Archived</a> from the original on 2021-12-11 – via YouTube.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Episode+4&rft.place=Sydney%2C+Australia&rft.series=WildLinAlg&rft.pub=University+of+New+South+Wales&rft.date=2010&rft.aulast=Wildberger&rft.aufirst=Norman+J.&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3D6XghF70fqkY&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://textbooks.math.gatech.edu/ila/determinants-volumes.html">"Determinants and Volumes"</a>. textbooks.math.gatech.edu. Retrieved 16 March 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=textbooks.math.gatech.edu&rft.atitle=Determinants+and+Volumes&rft_id=https%3A%2F%2Ftextbooks.math.gatech.edu%2Fila%2Fdeterminants-volumes.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcConnell1957" class="citation book cs1">McConnell (1957). <a rel="nofollow" class="external text" href="https://archive.org/details/applicationoften0000mcco">Applications of Tensor Analysis</a>. Dover Publications. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/applicationoften0000mcco/page/10">10–17</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Tensor+Analysis&rft.pages=10-17&rft.pub=Dover+Publications&rft.date=1957&rft.au=McConnell&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fapplicationoften0000mcco&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFHarris2014">Harris 2014</a>, §4.7 </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a>, Linear Algebra, 2nd Edition, Addison-Wesley, 1971, pp 173, 191. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFLang1987">Lang 1987</a>, §VI.7, Theorem 7.5 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Alternatively, <a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, §III.8, Proposition 1 proves this result using the <a href="/wiki/Functoriality" class="mw-redirect" title="Functoriality">functoriality</a> of the exterior power. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFHornJohnson2018">Horn & Johnson 2018</a>, §0.8.7 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFKungRotaYan2009">Kung, Rota & Yan 2009</a>, p. 306 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFHornJohnson2018">Horn & Johnson 2018</a>, §0.8.2. </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSilvester2000" class="citation journal cs1">Silvester, J. R. (2000). <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-01509379/document">"Determinants of Block Matrices"</a>. Math. Gaz. 84 (501): 460–467. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F3620776">10.2307/3620776</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/3620776">3620776</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:41879675">41879675</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Gaz.&rft.atitle=Determinants+of+Block+Matrices&rft.volume=84&rft.issue=501&rft.pages=460-467&rft.date=2000&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A41879675%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F3620776%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F3620776&rft.aulast=Silvester&rft.aufirst=J.+R.&rft_id=https%3A%2F%2Fhal.archives-ouvertes.fr%2Fhal-01509379%2Fdocument&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSothanaphan2017" class="citation journal cs1">Sothanaphan, Nat (January 2017). "Determinants of block matrices with noncommuting blocks". Linear Algebra and Its Applications. 512: 202–218. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1805.06027">1805.06027</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.laa.2016.10.004">10.1016/j.laa.2016.10.004</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119272194">119272194</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Linear+Algebra+and+Its+Applications&rft.atitle=Determinants+of+block+matrices+with+noncommuting+blocks&rft.volume=512&rft.pages=202-218&rft.date=2017-01&rft_id=info%3Aarxiv%2F1805.06027&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119272194%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.laa.2016.10.004&rft.aulast=Sothanaphan&rft.aufirst=Nat&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Proofs can be found in <a rel="nofollow" class="external free" href="http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html">http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html</a> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLinSra2014" class="citation arxiv cs1">Lin, Minghua; Sra, Suvrit (2014). "Completely strong superadditivity of generalized matrix functions". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1410.1958">1410.1958</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.FA">math.FA</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Completely+strong+superadditivity+of+generalized+matrix+functions&rft.date=2014&rft_id=info%3Aarxiv%2F1410.1958&rft.aulast=Lin&rft.aufirst=Minghua&rft.au=Sra%2C+Suvrit&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaksoyTurkmenZhang2014" class="citation journal cs1">Paksoy; Turkmen; Zhang (2014). <a rel="nofollow" class="external text" href="https://nsuworks.nova.edu/cgi/viewcontent.cgi?article=1062&context=math_facarticles">"Inequalities of Generalized Matrix Functions via Tensor Products"</a>. Electronic Journal of Linear Algebra. 27: 332–341. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.13001%2F1081-3810.1622">10.13001/1081-3810.1622</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Journal+of+Linear+Algebra&rft.atitle=Inequalities+of+Generalized+Matrix+Functions+via+Tensor+Products&rft.volume=27&rft.pages=332-341&rft.date=2014&rft_id=info%3Adoi%2F10.13001%2F1081-3810.1622&rft.au=Paksoy&rft.au=Turkmen&rft.au=Zhang&rft_id=https%3A%2F%2Fnsuworks.nova.edu%2Fcgi%2Fviewcontent.cgi%3Farticle%3D1062%26context%3Dmath_facarticles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre2010" class="citation web cs1">Serre, Denis (Oct 18, 2010). <a rel="nofollow" class="external text" href="https://mathoverflow.net/questions/42594/concavity-of-det1-n-over-hpd-n">"Concavity of det1/n over HPDn"</a>. MathOverflow.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathOverflow&rft.atitle=Concavity+of+det%3Csup%3E%3Csup%3E1%3C%2Fsup%3E%2F%3Csub%3En%3C%2Fsub%3E%3C%2Fsup%3E+over+HPD%3Csub%3En%3C%2Fsub%3E.&rft.date=2010-10-18&rft.aulast=Serre&rft.aufirst=Denis&rft_id=https%3A%2F%2Fmathoverflow.net%2Fquestions%2F42594%2Fconcavity-of-det1-n-over-hpd-n&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> <a href="#CITEREFLang1985">Lang 1985</a>, §VIII.2, <a href="#CITEREFHornJohnson2018">Horn & Johnson 2018</a>, Def. 1.2.3 </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFHornJohnson2018">Horn & Johnson 2018</a>, Observation 7.1.2, Theorem 7.2.5 </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> A proof can be found in the Appendix B of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKondratyukKrivoruchenko1992" class="citation journal cs1">Kondratyuk, L. A.; Krivoruchenko, M. I. (1992). "Superconducting quark matter in SU(2) color group". Zeitschrift für Physik A. 344 (1): 99–115. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992ZPhyA.344...99K">1992ZPhyA.344...99K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01291027">10.1007/BF01291027</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120467300">120467300</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Physik+A&rft.atitle=Superconducting+quark+matter+in+SU%282%29+color+group&rft.volume=344&rft.issue=1&rft.pages=99-115&rft.date=1992&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120467300%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01291027&rft_id=info%3Abibcode%2F1992ZPhyA.344...99K&rft.aulast=Kondratyuk&rft.aufirst=L.+A.&rft.au=Krivoruchenko%2C+M.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <a href="#CITEREFHornJohnson2018">Horn & Johnson 2018</a>, § 0.8.10 </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFGrattan-Guinness2003">Grattan-Guinness 2003</a>, §6.6 </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> Cajori, F. <a rel="nofollow" class="external text" href="https://archive.org/details/ahistorymathema02cajogoog/page/n94">A History of Mathematics p. 80</a> </li> <li id="cite_note-Campbell-24">^ <a href="#cite_ref-Campbell_24-0">a</a> <a href="#cite_ref-Campbell_24-1">b</a> <a href="#cite_ref-Campbell_24-2">c</a> Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFEves1990">Eves 1990</a>, p. 405 </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> A Brief History of Linear Algebra and Matrix Theory at: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120910034016/http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html">"A Brief History of Linear Algebra and Matrix Theory"</a>. Archived from <a rel="nofollow" class="external text" href="http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html">the original</a> on 10 September 2012. Retrieved 24 January 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Brief+History+of+Linear+Algebra+and+Matrix+Theory&rft_id=http%3A%2F%2Fdarkwing.uoregon.edu%2F~vitulli%2F441.sp04%2FLinAlgHistory.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFKleiner2007">Kleiner 2007</a>, p. 80 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="#CITEREFBourbaki1994">Bourbaki (1994</a>, p. 59) </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> Muir, Sir Thomas, The Theory of Determinants in the historical Order of Development [London, England: Macmillan and Co., Ltd., 1906]. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:37.0181.02">37.0181.02</a> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <a href="#CITEREFKleiner2007">Kleiner 2007</a>, §5.2 </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> The first use of the word "determinant" in the modern sense appeared in: Cauchy, Augustin-Louis "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment," which was first read at the Institute de France in Paris on November 30, 1812, and which was subsequently published in the Journal de l'Ecole Polytechnique, Cahier 17, Tome 10, pages 29–112 (1815). </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Origins of mathematical terms: <a rel="nofollow" class="external free" href="http://jeff560.tripod.com/d.html">http://jeff560.tripod.com/d.html</a> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> History of matrices and determinants: <a rel="nofollow" class="external free" href="http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html">http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html</a> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <a href="#CITEREFEves1990">Eves 1990</a>, p. 494 </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <a href="#CITEREFCajori1993">Cajori 1993</a>, Vol. II, p. 92, no. 462 </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> History of matrix notation: <a rel="nofollow" class="external free" href="http://jeff560.tripod.com/matrices.html">http://jeff560.tripod.com/matrices.html</a> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="#CITEREFHabgoodArel2012">Habgood & Arel 2012</a> </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <a href="#CITEREFLang1985">Lang 1985</a>, §VII.3 </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, §IV.8 </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <a href="#CITEREFLang1985">Lang 1985</a>, §VII.6, Theorem 6.10 </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLay2021" class="citation book cs1">Lay, David (2021). Linear Algebra and Its Applications 6th Edition. Pearson. p. 172.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications+6th+Edition&rft.pages=172&rft.pub=Pearson&rft.date=2021&rft.aulast=Lay&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote 2004</a>, §11.4 </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote 2004</a>, §11.4, Theorem 30 </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <a href="#CITEREFMac_Lane1998">Mac Lane 1998</a>, §I.4. See also <a href="/wiki/Natural_transformation#Determinant" title="Natural transformation">Natural transformation § Determinant</a>. </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, §III.8 </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <a href="#CITEREFLombardiQuitté2015">Lombardi & Quitté 2015</a>, §5.2, <a href="#CITEREFBourbaki1998">Bourbaki 1998</a>, §III.5 </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <a href="#CITEREFGaribaldi2004">Garibaldi 2004</a> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalars a, b: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}ab&=ab{\begin{vmatrix}1&0\\0&1\end{vmatrix}}=a{\begin{vmatrix}1&0\\0&b\end{vmatrix}}\\[5mu]&={\begin{vmatrix}a&0\\0&b\end{vmatrix}}=b{\begin{vmatrix}a&0\\0&1\end{vmatrix}}=ba{\begin{vmatrix}1&0\\0&1\end{vmatrix}}=ba,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.578em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}ab&=ab{\begin{vmatrix}1&0\\0&1\end{vmatrix}}=a{\begin{vmatrix}1&0\\0&b\end{vmatrix}}\\[5mu]&={\begin{vmatrix}a&0\\0&b\end{vmatrix}}=b{\begin{vmatrix}a&0\\0&1\end{vmatrix}}=ba{\begin{vmatrix}1&0\\0&1\end{vmatrix}}=ba,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/157e9651bed4fba99a1bb699ce9d126b7fee0440" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:41.684ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}ab&=ab{\begin{vmatrix}1&0\\0&1\end{vmatrix}}=a{\begin{vmatrix}1&0\\0&b\end{vmatrix}}\\[5mu]&={\begin{vmatrix}a&0\\0&b\end{vmatrix}}=b{\begin{vmatrix}a&0\\0&1\end{vmatrix}}=ba{\begin{vmatrix}1&0\\0&1\end{vmatrix}}=ba,\end{aligned}}}"> a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVaradarajan2004" class="citation cs2">Varadarajan, V. S (2004), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sZ1-G4hQgIIC&q=Berezinian&pg=PA116">Supersymmetry for mathematicians: An introduction</a>, American Mathematical Soc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3574-6" title="Special:BookSources/978-0-8218-3574-6"><bdi>978-0-8218-3574-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Supersymmetry+for+mathematicians%3A+An+introduction&rft.pub=American+Mathematical+Soc.&rft.date=2004&rft.isbn=978-0-8218-3574-6&rft.aulast=Varadarajan&rft.aufirst=V.+S&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsZ1-G4hQgIIC%26q%3DBerezinian%26pg%3DPA116&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> "... we mention that the determinant, though a convenient notion theoretically, rarely finds a useful role in numerical algorithms.", see <a href="#CITEREFTrefethenBau_III1997">Trefethen & Bau III 1997</a>, Lecture 1. </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <a href="#CITEREFFisikopoulosPeñaranda2016">Fisikopoulos & Peñaranda 2016</a>, §1.1, §4.3 </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCamarero2018" class="citation arxiv cs1">Camarero, Cristóbal (2018-12-05). "Simple, Fast and Practicable Algorithms for Cholesky, LU and QR Decomposition Using Fast Rectangular Matrix Multiplication". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1812.02056">1812.02056</a> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.NA">cs.NA</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Simple%2C+Fast+and+Practicable+Algorithms+for+Cholesky%2C+LU+and+QR+Decomposition+Using+Fast+Rectangular+Matrix+Multiplication&rft.date=2018-12-05&rft_id=info%3Aarxiv%2F1812.02056&rft.aulast=Camarero&rft.aufirst=Crist%C3%B3bal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <a href="#CITEREFBunchHopcroft1974">Bunch & Hopcroft 1974</a> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <a href="#CITEREFFisikopoulosPeñaranda2016">Fisikopoulos & Peñaranda 2016</a>, §1.1 </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <a href="#CITEREFRote2001">Rote 2001</a> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFangHavas1997" class="citation conference cs1">Fang, Xin Gui; Havas, George (1997). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110807042828/http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/PDF/ft_gateway.cfm.pdf">"On the worst-case complexity of integer Gaussian elimination"</a> (PDF). Proceedings of the 1997 international symposium on Symbolic and algebraic computation. ISSAC '97. Kihei, Maui, Hawaii, United States: ACM. pp. 28–31. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F258726.258740">10.1145/258726.258740</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89791-875-4" title="Special:BookSources/0-89791-875-4"><bdi>0-89791-875-4</bdi></a>. Archived from <a rel="nofollow" class="external text" href="http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/PDF/ft_gateway.cfm.pdf">the original</a> (PDF) on 2011-08-07. Retrieved 2011-01-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=On+the+worst-case+complexity+of+integer+Gaussian+elimination&rft.btitle=Proceedings+of+the+1997+international+symposium+on+Symbolic+and+algebraic+computation&rft.place=Kihei%2C+Maui%2C+Hawaii%2C+United+States&rft.pages=28-31&rft.pub=ACM&rft.date=1997&rft_id=info%3Adoi%2F10.1145%2F258726.258740&rft.isbn=0-89791-875-4&rft.aulast=Fang&rft.aufirst=Xin+Gui&rft.au=Havas%2C+George&rft_id=http%3A%2F%2Fperso.ens-lyon.fr%2Fgilles.villard%2FBIBLIOGRAPHIE%2FPDF%2Fft_gateway.cfm.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <a href="#CITEREFFisikopoulosPeñaranda2016">Fisikopoulos & Peñaranda 2016</a>, §1.1, <a href="#CITEREFBareiss1968">Bareiss 1968</a> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAbeles2008" class="citation journal cs1">Abeles, Francine F. (2008). <a rel="nofollow" class="external text" href="https://www.academia.edu/10352246">"Dodgson condensation: The historical and mathematical development of an experimental method"</a>. Linear Algebra and Its Applications. 429 (2–3): 429–438. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.laa.2007.11.022">10.1016/j.laa.2007.11.022</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Linear+Algebra+and+Its+Applications&rft.atitle=Dodgson+condensation%3A+The+historical+and+mathematical+development+of+an+experimental+method&rft.volume=429&rft.issue=2%E2%80%933&rft.pages=429-438&rft.date=2008&rft_id=info%3Adoi%2F10.1016%2Fj.laa.2007.11.022&rft.aulast=Abeles&rft.aufirst=Francine+F.&rft_id=https%3A%2F%2Fwww.academia.edu%2F10352246&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=46" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Linear_algebra#Further_reading" title="Linear algebra">Linear algebra § Further reading</a></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnton2005" class="citation cs2">Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Linear+Algebra+%28Applications+Version%29&rft.edition=9th&rft.pub=Wiley+International&rft.date=2005&rft.aulast=Anton&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAxler2015" class="citation book cs1"><a href="/wiki/Sheldon_Axler" title="Sheldon Axler">Axler, Sheldon Jay</a> (2015). Linear Algebra Done Right (3rd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-11079-0" title="Special:BookSources/978-3-319-11079-0"><bdi>978-3-319-11079-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+Done+Right&rft.edition=3rd&rft.pub=Springer&rft.date=2015&rft.isbn=978-3-319-11079-0&rft.aulast=Axler&rft.aufirst=Sheldon+Jay&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBareiss1968" class="citation cs2">Bareiss, Erwin (1968), <a rel="nofollow" class="external text" href="https://www.ams.org/journals/mcom/1968-22-103/S0025-5718-1968-0226829-0/S0025-5718-1968-0226829-0.pdf">"Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination"</a> (PDF), Mathematics of Computation, 22 (102): 565–578, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2004533">10.2307/2004533</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2004533">2004533</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20121025053848/http://www.ams.org/journals/mcom/1968-22-103/S0025-5718-1968-0226829-0/S0025-5718-1968-0226829-0.pdf">archived</a> (PDF) from the original on 2012-10-25</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+of+Computation&rft.atitle=Sylvester%27s+Identity+and+Multistep+Integer-Preserving+Gaussian+Elimination&rft.volume=22&rft.issue=102&rft.pages=565-578&rft.date=1968&rft_id=info%3Adoi%2F10.2307%2F2004533&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2004533%23id-name%3DJSTOR&rft.aulast=Bareiss&rft.aufirst=Erwin&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fmcom%2F1968-22-103%2FS0025-5718-1968-0226829-0%2FS0025-5718-1968-0226829-0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFde_Boor1990" class="citation cs2"><a href="/wiki/Carl_R._de_Boor" title="Carl R. de Boor">de Boor, Carl</a> (1990), <a rel="nofollow" class="external text" href="http://ftp.cs.wisc.edu/Approx/empty.pdf">"An empty exercise"</a> (PDF), ACM SIGNUM Newsletter, 25 (2): 3–7, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F122272.122273">10.1145/122272.122273</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:62780452">62780452</a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060901214854/http://ftp.cs.wisc.edu/Approx/empty.pdf">archived</a> (PDF) from the original on 2006-09-01</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=ACM+SIGNUM+Newsletter&rft.atitle=An+empty+exercise&rft.volume=25&rft.issue=2&rft.pages=3-7&rft.date=1990&rft_id=info%3Adoi%2F10.1145%2F122272.122273&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A62780452%23id-name%3DS2CID&rft.aulast=de+Boor&rft.aufirst=Carl&rft_id=http%3A%2F%2Fftp.cs.wisc.edu%2FApprox%2Fempty.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1998" class="citation cs2">Bourbaki, Nicolas (1998), Algebra I, Chapters 1-3, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540642435" title="Special:BookSources/9783540642435"><bdi>9783540642435</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+I%2C+Chapters+1-3&rft.pub=Springer&rft.date=1998&rft.isbn=9783540642435&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBunchHopcroft1974" class="citation journal cs1">Bunch, J. 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Journal of Discrete Algorithms. 10: 98–109. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jda.2011.06.007">10.1016/j.jda.2011.06.007</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190505060158/https://hal.archives-ouvertes.fr/hal-01500199/file/HA.pdf">Archived</a> (PDF) from the original on 2019-05-05.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Discrete+Algorithms&rft.atitle=A+condensation-based+application+of+Cramer%27s+rule+for+solving+large-scale+linear+systems&rft.volume=10&rft.pages=98-109&rft.date=2012&rft_id=info%3Adoi%2F10.1016%2Fj.jda.2011.06.007&rft.aulast=Habgood&rft.aufirst=Ken&rft.au=Arel%2C+Itamar&rft_id=https%3A%2F%2Fhal.archives-ouvertes.fr%2Fhal-01500199%2Ffile%2FHA.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarris2014" class="citation cs2">Harris, Frank E. (2014), Mathematics for Physical Science and Engineering, Elsevier, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780128010495" title="Special:BookSources/9780128010495"><bdi>9780128010495</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+for+Physical+Science+and+Engineering&rft.pub=Elsevier&rft.date=2014&rft.isbn=9780128010495&rft.aulast=Harris&rft.aufirst=Frank+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner2007" class="citation cs2">Kleiner, Israel (2007), Kleiner, Israel (ed.), A history of abstract algebra, Birkhäuser, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-0-8176-4685-1">10.1007/978-0-8176-4685-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4684-4" title="Special:BookSources/978-0-8176-4684-4"><bdi>978-0-8176-4684-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2347309">2347309</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+abstract+algebra&rft.pub=Birkh%C3%A4user&rft.date=2007&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2347309%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-0-8176-4685-1&rft.isbn=978-0-8176-4684-4&rft.aulast=Kleiner&rft.aufirst=Israel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKungRotaYan2009" class="citation cs2">Kung, Joseph P.S.; Rota, Gian-Carlo; <a href="/wiki/Catherine_H._Yan" class="mw-redirect" title="Catherine H. 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Metzler, New York, NY: Dover</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+treatise+on+the+theory+of+determinants&rft.place=New+York%2C+NY&rft.pub=Dover&rft.date=1960&rft.aulast=Muir&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoole2006" class="citation cs2">Poole, David (2006), Linear Algebra: A Modern Introduction (2nd ed.), Brooks/Cole, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-534-99845-3" title="Special:BookSources/0-534-99845-3"><bdi>0-534-99845-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra%3A+A+Modern+Introduction&rft.edition=2nd&rft.pub=Brooks%2FCole&rft.date=2006&rft.isbn=0-534-99845-3&rft.aulast=Poole&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><a href="/wiki/G._Baley_Price" class="mw-redirect" title="G. Baley Price">G. Baley Price</a> (1947) "Some identities in the theory of determinants", <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a> 54:75–90 <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0019078">0019078</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson2018" class="citation book cs1"><a href="/wiki/Roger_Horn" title="Roger Horn">Horn, Roger Alan</a>; <a href="/wiki/Charles_Royal_Johnson" title="Charles Royal Johnson">Johnson, Charles Royal</a> (2018) [1985]. Matrix Analysis (2nd ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-54823-6" title="Special:BookSources/978-0-521-54823-6"><bdi>978-0-521-54823-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2018&rft.isbn=978-0-521-54823-6&rft.aulast=Horn&rft.aufirst=Roger+Alan&rft.au=Johnson%2C+Charles+Royal&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1985" class="citation cs2">Lang, Serge (1985), Introduction to Linear Algebra, Undergraduate Texts in Mathematics (2 ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387962054" title="Special:BookSources/9780387962054"><bdi>9780387962054</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Linear+Algebra&rft.series=Undergraduate+Texts+in+Mathematics&rft.edition=2&rft.pub=Springer&rft.date=1985&rft.isbn=9780387962054&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1987" class="citation cs2">Lang, Serge (1987), Linear Algebra, Undergraduate Texts in Mathematics (3 ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387964126" title="Special:BookSources/9780387964126"><bdi>9780387964126</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.series=Undergraduate+Texts+in+Mathematics&rft.edition=3&rft.pub=Springer&rft.date=1987&rft.isbn=9780387964126&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2002" class="citation book cs1">Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. New York, NY: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95385-4" title="Special:BookSources/978-0-387-95385-4"><bdi>978-0-387-95385-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.place=New+York%2C+NY&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2002&rft.isbn=978-0-387-95385-4&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLeon2006" class="citation cs2">Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+With+Applications&rft.edition=7th&rft.pub=Pearson+Prentice+Hall&rft.date=2006&rft.aulast=Leon&rft.aufirst=Steven+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRote2001" class="citation cs2">Rote, Günter (2001), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070201145100/http://page.inf.fu-berlin.de/~rote/Papers/pdf/Division-free+algorithms.pdf">"Division-free algorithms for the determinant and the Pfaffian: algebraic and combinatorial approaches"</a> (PDF), Computational discrete mathematics, Lecture Notes in Comput. Sci., vol. 2122, Springer, pp. 119–135, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-45506-X_9">10.1007/3-540-45506-X_9</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-42775-9" title="Special:BookSources/978-3-540-42775-9"><bdi>978-3-540-42775-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1911585">1911585</a>, archived from <a rel="nofollow" class="external text" href="https://page.inf.fu-berlin.de/~rote/Papers/pdf/Division-free+algorithms.pdf">the original</a> (PDF) on 2007-02-01, retrieved 2020-06-04</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Division-free+algorithms+for+the+determinant+and+the+Pfaffian%3A+algebraic+and+combinatorial+approaches&rft.btitle=Computational+discrete+mathematics&rft.series=Lecture+Notes+in+Comput.+Sci.&rft.pages=119-135&rft.pub=Springer&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1911585%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F3-540-45506-X_9&rft.isbn=978-3-540-42775-9&rft.aulast=Rote&rft.aufirst=G%C3%BCnter&rft_id=https%3A%2F%2Fpage.inf.fu-berlin.de%2F~rote%2FPapers%2Fpdf%2FDivision-free%2Balgorithms.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrefethenBau_III1997" class="citation cs2">Trefethen, Lloyd; Bau III, David (1997), Numerical Linear Algebra (1st ed.), Philadelphia: SIAM, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89871-361-9" title="Special:BookSources/978-0-89871-361-9"><bdi>978-0-89871-361-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Linear+Algebra&rft.place=Philadelphia&rft.edition=1st&rft.pub=SIAM&rft.date=1997&rft.isbn=978-0-89871-361-9&rft.aulast=Trefethen&rft.aufirst=Lloyd&rft.au=Bau+III%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Historical_references">Historical references</h3>[<a href="/w/index.php?title=Determinant&action=edit&section=47" title="Edit section: Historical references">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1994" class="citation cs2">Bourbaki, Nicolas (1994), Elements of the history of mathematics, translated by <a href="/wiki/John_D._P._Meldrum" title="John D. P. Meldrum">Meldrum, John</a>, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-61693-8">10.1007/978-3-642-61693-8</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-19376-6" title="Special:BookSources/3-540-19376-6"><bdi>3-540-19376-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+the+history+of+mathematics&rft.pub=Springer&rft.date=1994&rft_id=info%3Adoi%2F10.1007%2F978-3-642-61693-8&rft.isbn=3-540-19376-6&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCajori1993" class="citation cs2">Cajori, Florian (1993), A history of mathematical notations: Including Vol. I. Notations in elementary mathematics; Vol. II. Notations mainly in higher mathematics, Reprint of the 1928 and 1929 originals, Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-67766-4" title="Special:BookSources/0-486-67766-4"><bdi>0-486-67766-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3363427">3363427</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+history+of+mathematical+notations%3A+Including+Vol.+I.+Notations+in+elementary+mathematics%3B+Vol.+II.+Notations+mainly+in+higher+mathematics%2C+Reprint+of+the+1928+and+1929+originals&rft.pub=Dover&rft.date=1993&rft.isbn=0-486-67766-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3363427%23id-name%3DMR&rft.aulast=Cajori&rft.aufirst=Florian&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBézout1779" class="citation cs2">Bézout, Étienne (1779), <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k106053p.image">Théorie générale des equations algébriques</a>, Paris</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+g%C3%A9n%C3%A9rale+des+equations+alg%C3%A9briques&rft.place=Paris&rft.date=1779&rft.aulast=B%C3%A9zout&rft.aufirst=%C3%89tienne&rft_id=https%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k106053p.image&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCayley1841" class="citation cs2">Cayley, Arthur (1841), "On a theorem in the geometry of position", Cambridge Mathematical Journal, 2: 267–271</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Cambridge+Mathematical+Journal&rft.atitle=On+a+theorem+in+the+geometry+of+position&rft.volume=2&rft.pages=267-271&rft.date=1841&rft.aulast=Cayley&rft.aufirst=Arthur&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCramer1750" class="citation cs2">Cramer, Gabriel (1750), Introduction à l'analyse des lignes courbes algébriques, Genève: Frères Cramer & Cl. Philibert, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3931%2Fe-rara-4048">10.3931/e-rara-4048</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+%C3%A0+l%27analyse+des+lignes+courbes+alg%C3%A9briques&rft.place=Gen%C3%A8ve&rft.pub=Fr%C3%A8res+Cramer+%26+Cl.+Philibert&rft.date=1750&rft_id=info%3Adoi%2F10.3931%2Fe-rara-4048&rft.aulast=Cramer&rft.aufirst=Gabriel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1990" class="citation cs2">Eves, Howard (1990), An introduction to the history of mathematics (6 ed.), Saunders College Publishing, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-029558-0" title="Special:BookSources/0-03-029558-0"><bdi>0-03-029558-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1104435">1104435</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+history+of+mathematics&rft.edition=6&rft.pub=Saunders+College+Publishing&rft.date=1990&rft.isbn=0-03-029558-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1104435%23id-name%3DMR&rft.aulast=Eves&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrattan-Guinness2003" class="citation cs2">Grattan-Guinness, I., ed. (2003), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, vol. 1, <a href="/wiki/Johns_Hopkins_University_Press" title="Johns Hopkins University Press">Johns Hopkins University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780801873966" title="Special:BookSources/9780801873966"><bdi>9780801873966</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Companion+Encyclopedia+of+the+History+and+Philosophy+of+the+Mathematical+Sciences&rft.pub=Johns+Hopkins+University+Press&rft.date=2003&rft.isbn=9780801873966&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobi1841" class="citation cs2"><a href="/wiki/Carl_Gustav_Jakob_Jacobi" class="mw-redirect" title="Carl Gustav Jakob Jacobi">Jacobi, Carl Gustav Jakob</a> (1841), <a rel="nofollow" class="external text" href="https://www.digizeitschriften.de/dms/img/?PID=GDZPPN002142724&physid=phys325#navi">"De Determinantibus functionalibus"</a>, Journal für die reine und angewandte Mathematik, 1841 (22): 320–359, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1841.22.319">10.1515/crll.1841.22.319</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123637858">123637858</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=De+Determinantibus+functionalibus&rft.volume=1841&rft.issue=22&rft.pages=320-359&rft.date=1841&rft_id=info%3Adoi%2F10.1515%2Fcrll.1841.22.319&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123637858%23id-name%3DS2CID&rft.aulast=Jacobi&rft.aufirst=Carl+Gustav+Jakob&rft_id=https%3A%2F%2Fwww.digizeitschriften.de%2Fdms%2Fimg%2F%3FPID%3DGDZPPN002142724%26physid%3Dphys325%23navi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaplace1772" class="citation cs2"><a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace, Pierre-Simon, de</a> (1772), <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k77596b/f374">"Recherches sur le calcul intégral et sur le systéme du monde"</a>, Histoire de l'Académie Royale des Sciences (seconde partie), Paris: 267–376</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Histoire+de+l%27Acad%C3%A9mie+Royale+des+Sciences&rft.atitle=Recherches+sur+le+calcul+int%C3%A9gral+et+sur+le+syst%C3%A9me+du+monde&rft.issue=seconde+partie&rft.pages=267-376&rft.date=1772&rft.aulast=Laplace&rft.aufirst=Pierre-Simon%2C+de&rft_id=https%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k77596b%2Ff374&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Citation" title="Template:Citation">citation</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Determinant&action=edit&section=48" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></div> <div class="side-box-text plainlist">The Wikibook <a href="https://en.wikibooks.org/wiki/Linear_Algebra" class="extiw" title="wikibooks:Linear Algebra">Linear Algebra</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Linear_Algebra/Determinants" class="extiw" title="wikibooks:Linear Algebra/Determinants">Determinants</a></div></div> </div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/38px-Wikisource-logo.svg.png" decoding="async" width="38" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/57px-Wikisource-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/76px-Wikisource-logo.svg.png 2x" data-file-width="410" data-file-height="430" /></div> <div class="side-box-text plainlist"><a href="/wiki/Wikisource" title="Wikisource">Wikisource</a> has the text of the <a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">1911 Encyclopædia Britannica</a> article "<a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Determinant" class="extiw" title="wikisource:1911 Encyclopædia Britannica/Determinant">Determinant</a>".</div></div> </div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuprunenko2001" class="citation cs2">Suprunenko, D.A. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Determinant">"Determinant"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Determinant&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Suprunenko&rft.aufirst=D.A.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DDeterminant&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Determinant.html">"Determinant"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Determinant&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FDeterminant.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Matrices_and_determinants.html">"Matrices and determinants"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Matrices+and+determinants&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FMatrices_and_determinants.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADeterminant" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://people.revoledu.com/kardi/tutorial/LinearAlgebra/MatrixDeterminant.html">Determinant Interactive Program and Tutorial</a></li> <li><a rel="nofollow" class="external text" href="http://www.umat.feec.vutbr.cz/~novakm/determinanty/en/">Linear algebra: determinants.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081204081902/http://www.umat.feec.vutbr.cz/~novakm/determinanty/en/">Archived</a> 2008-12-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> Compute determinants of matrices up to order 6 using Laplace expansion you choose.</li> <li><a rel="nofollow" class="external text" href="https://physandmathsolutions.com/Menus/matrix_determinant_calculator.php">Determinant Calculator</a> Calculator for matrix determinants, up to the 8th order.</li> <li><a rel="nofollow" class="external text" href="http://www.economics.soton.ac.uk/staff/aldrich/matrices.htm">Matrices and Linear Algebra on the Earliest Uses Pages</a></li> <li><a rel="nofollow" class="external text" href="http://algebra.math.ust.hk/course/content.shtml">Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course.</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist 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style="font-size:114%;margin:0 4em"><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Outline_of_linear_algebra" title="Outline of linear algebra">Outline</a></li> <li><a href="/wiki/Glossary_of_linear_algebra" title="Glossary of linear algebra">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">Scalar</a></li> <li><a href="/wiki/Euclidean_vector" title="Euclidean vector">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li> <li><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></li> <li><a href="/wiki/Vector_projection" title="Vector projection">Vector projection</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Linear projection</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a></li> <li><a href="/wiki/Multilinear_map" title="Multilinear map">Multilinear map</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Change_of_basis" title="Change of basis">Change of basis</a></li> <li><a href="/wiki/Row_and_column_vectors" title="Row and column vectors">Row and column vectors</a></li> <li><a href="/wiki/Row_and_column_spaces" title="Row and column spaces">Row and column spaces</a></li> <li><a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a></li> <li><a href="/wiki/System_of_linear_equations" title="System of linear equations">Linear equations</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, 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href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li> <li><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></li> <li><a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li> <li><a href="/wiki/Productive_matrix" title="Productive matrix">Productive matrix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a></li> <li><a href="/wiki/Dot_product" title="Dot product">Dot product</a></li> <li><a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Outer_product" title="Outer product">Outer product</a></li> <li><a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a></li> <li><a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Determinant</a></li> <li><a href="/wiki/Cross_product" title="Cross product">Cross product</a></li> <li><a href="/wiki/Triple_product" title="Triple product">Triple product</a></li> <li><a href="/wiki/Seven-dimensional_cross_product" title="Seven-dimensional cross product">Seven-dimensional cross product</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Bivector" title="Bivector">Bivector</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Outermorphism" title="Outermorphism">Outermorphism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_space" title="Vector space">Vector space</a> constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_space" title="Dual space">Dual</a></li> <li><a href="/wiki/Direct_sum_of_modules#Construction_for_two_vector_spaces" title="Direct sum of modules">Direct sum</a></li> <li><a href="/wiki/Function_space#In_linear_algebra" title="Function space">Function space</a></li> <li><a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">Quotient</a></li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Subspace</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a></li> <li><a href="/wiki/Numerical_stability" title="Numerical stability">Numerical stability</a></li> <li><a href="/wiki/Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">Basic Linear Algebra Subprograms</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li> <li><a 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Template:Navbox"," 8.13% 112.889 2 Template:Pagetype"," 7.62% 105.918 1 Template:Cite_AV_media"," 5.86% 81.382 68 Template:Math"," 4.96% 68.855 5 Template:Harvtxt"]},"scribunto":{"limitreport-timeusage":{"value":"0.870","limit":"10.000"},"limitreport-memusage":{"value":8680824,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"Abstract_formulation\"] = 1,\n [\"CITEREFAbeles2008\"] = 1,\n [\"CITEREFAnton2005\"] = 1,\n [\"CITEREFAxler2015\"] = 1,\n [\"CITEREFBareiss1968\"] = 1,\n [\"CITEREFBourbaki1994\"] = 1,\n [\"CITEREFBourbaki1998\"] = 1,\n [\"CITEREFBunchHopcroft1974\"] = 1,\n [\"CITEREFBézout1779\"] = 1,\n [\"CITEREFCajori1993\"] = 1,\n [\"CITEREFCamarero2018\"] = 1,\n [\"CITEREFCayley1841\"] = 1,\n [\"CITEREFCramer1750\"] = 1,\n [\"CITEREFDummitFoote2004\"] = 1,\n [\"CITEREFEves1990\"] = 1,\n [\"CITEREFFangHavas1997\"] = 1,\n [\"CITEREFFisikopoulosPeñaranda2016\"] = 1,\n [\"CITEREFGaribaldi2004\"] = 1,\n [\"CITEREFGrattan-Guinness2003\"] = 1,\n [\"CITEREFHabgoodArel2012\"] = 1,\n [\"CITEREFHarris2014\"] = 1,\n [\"CITEREFHornJohnson2018\"] = 1,\n [\"CITEREFJacobi1841\"] = 1,\n [\"CITEREFKleiner2007\"] = 1,\n [\"CITEREFKondratyukKrivoruchenko1992\"] = 1,\n [\"CITEREFKungRotaYan2009\"] = 1,\n [\"CITEREFLang1985\"] = 1,\n [\"CITEREFLang1987\"] = 1,\n [\"CITEREFLang2002\"] = 1,\n [\"CITEREFLaplace1772\"] = 1,\n [\"CITEREFLay2005\"] = 1,\n [\"CITEREFLay2021\"] = 1,\n [\"CITEREFLeon2006\"] = 1,\n [\"CITEREFLinSra2014\"] = 1,\n [\"CITEREFLombardiQuitté2015\"] = 1,\n [\"CITEREFMac_Lane1998\"] = 1,\n [\"CITEREFMcConnell1957\"] = 1,\n [\"CITEREFMeyer2001\"] = 1,\n [\"CITEREFMuir1960\"] = 1,\n [\"CITEREFPaksoyTurkmenZhang2014\"] = 1,\n [\"CITEREFPoole2006\"] = 1,\n [\"CITEREFRote2001\"] = 1,\n [\"CITEREFSerre2010\"] = 1,\n [\"CITEREFSilvester2000\"] = 1,\n [\"CITEREFSothanaphan2017\"] = 1,\n [\"CITEREFTrefethenBau_III1997\"] = 1,\n [\"CITEREFVaradarajan2004\"] = 1,\n [\"CITEREFWildberger2010\"] = 1,\n [\"CITEREFde_Boor1990\"] = 1,\n}\ntemplate_list = table#1 {\n [\"!\"] = 8,\n [\"=\"] = 1,\n [\"About\"] = 1,\n [\"Abs\"] = 1,\n [\"Anchor\"] = 1,\n [\"Authority control\"] = 1,\n [\"Cbignore\"] = 1,\n [\"Citation\"] = 31,\n [\"Citation needed\"] = 2,\n [\"Cite AV media\"] = 1,\n [\"Cite arXiv\"] = 2,\n [\"Cite book\"] = 5,\n [\"Cite conference\"] = 1,\n [\"Cite journal\"] = 7,\n [\"Cite web\"] = 3,\n [\"Clarify\"] = 1,\n [\"Clarify span\"] = 1,\n [\"Colbegin\"] = 1,\n [\"Colend\"] = 1,\n [\"EB1911 poster\"] = 1,\n [\"Harv\"] = 1,\n [\"Harvnb\"] = 36,\n [\"Harvtxt\"] = 5,\n [\"JFM\"] = 1,\n [\"Linear algebra\"] = 1,\n [\"MacTutor\"] = 1,\n [\"Main\"] = 2,\n [\"Math\"] = 68,\n [\"MathWorld\"] = 1,\n [\"Mr\"] = 1,\n [\"Mvar\"] = 31,\n [\"Ordered list\"] = 1,\n [\"Portal\"] = 1,\n [\"Reflist\"] = 2,\n [\"Section link\"] = 1,\n [\"See also\"] = 2,\n [\"Short description\"] = 1,\n [\"SpringerEOM\"] = 1,\n [\"Webarchive\"] = 1,\n [\"Wikibooks\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-szq4t","timestamp":"20241122140559","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Determinant","url":"https:\/\/en.wikipedia.org\/wiki\/Determinant","sameAs":"http:\/\/www.wikidata.org\/entity\/Q178546","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q178546","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-09-14T21:11:50Z","dateModified":"2024-11-19T20:19:04Z","headline":"sum of signed terms of n factors from n\u00d7n matrix with no two factors sharing row or column"}</script> </body> </html>

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