Dodgson condensation

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</div> </header> <main id="content" class="mw-body"> <div class="banner-container"> <div id="siteNotice"></div> </div> <div class="pre-content heading-holder"> <div class="page-heading"> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Dodgson condensation</span></h1> <div class="tagline"></div> </div> <ul id="p-associated-pages" class="minerva__tab-container"> <li class="minerva__tab selected"> <a class="minerva__tab-text" href="/wiki/Dodgson_condensation" rel="" data-event-name="tabs.subject">Article</a> </li> <li class="minerva__tab "> <a class="minerva__tab-text" href="/wiki/Talk:Dodgson_condensation" rel="discussion" data-event-name="tabs.talk">Talk</a> </li> </ul> <nav class="page-actions-menu"> <ul id="p-views" class="page-actions-menu__list"> <li id="language-selector" class="page-actions-menu__list-item"> <a role="button" href="#p-lang" data-mw="interface" data-event-name="menu.languages" title="Language" class="cdx-button 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cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> <span class="minerva-icon minerva-icon--edit"></span> <span>Edit</span> </a> </li> </ul> </nav>  <div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Lewis_Carroll_identity&redirect=no" class="mw-redirect" title="Lewis Carroll identity">Lewis Carroll identity</a>)</span></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"><script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><section class="mf-section-0" id="mf-section-0"> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">January 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>Dodgson condensation</b> or <b>method of contractants</b> is a method of computing the <a href="/wiki/Determinant" title="Determinant">determinants</a> of <a href="/wiki/Square_matrix" title="Square matrix">square matrices</a>. It is named for its inventor, <a href="/wiki/Charles_Lutwidge_Dodgson" class="mw-redirect" title="Charles Lutwidge Dodgson">Charles Lutwidge Dodgson</a> (better known by his pseudonym, as Lewis Carroll, the popular author), who discovered it in 1866.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The method in the case of an <i>n</i> × <i>n</i> matrix is to construct an (<i>n</i> − 1) × (<i>n</i> − 1) matrix, an (<i>n</i> − 2) × (<i>n</i> − 2), and so on, finishing with a 1 × 1 matrix, which has one entry, the determinant of the original matrix. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#General_method"><span class="tocnumber">1</span> <span class="toctext">General method</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Examples"><span class="tocnumber">2</span> <span class="toctext">Examples</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Without_zeros"><span class="tocnumber">2.1</span> <span class="toctext">Without zeros</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#With_zeros"><span class="tocnumber">2.2</span> <span class="toctext">With zeros</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-5"><a href="#Desnanot%E2%80%93Jacobi_identity_and_proof_of_correctness_of_the_condensation_algorithm"><span class="tocnumber">3</span> <span class="toctext">Desnanot–Jacobi identity and proof of correctness of the condensation algorithm</span></a> <ul> <li class="toclevel-2 tocsection-6"><a href="#Desnanot%E2%80%93Jacobi_identity"><span class="tocnumber">3.1</span> <span class="toctext">Desnanot–Jacobi identity</span></a></li> <li class="toclevel-2 tocsection-7"><a href="#Proof_of_the_correctness_of_Dodgson_condensation"><span class="tocnumber">3.2</span> <span class="toctext">Proof of the correctness of Dodgson condensation</span></a></li> <li class="toclevel-2 tocsection-8"><a href="#Proof_of_the_Desnanot-Jacobi_identity"><span class="tocnumber">3.3</span> <span class="toctext">Proof of the Desnanot-Jacobi identity</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-9"><a href="#References"><span class="tocnumber">4</span> <span class="toctext">References</span></a></li> <li class="toclevel-1 tocsection-10"><a href="#Further_reading"><span class="tocnumber">5</span> <span class="toctext">Further reading</span></a></li> <li class="toclevel-1 tocsection-11"><a href="#External_links"><span class="tocnumber">6</span> <span class="toctext">External links</span></a></li> </ul> </div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="General_method">General method</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=1" title="Edit section: General method" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-1 collapsible-block" id="mf-section-1"> <p>This algorithm can be described in the following four steps: </p> <ol><li>Let A be the given <i>n</i> × <i>n</i> matrix. Arrange A so that no zeros occur in its interior. An explicit definition of interior would be all a<sub>i,j</sub> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j\neq 1,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≠</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i,j\neq 1,n}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875dc47ba5de13bbbafbfa58a3c9dbc029b1ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.484ex; height:2.676ex;" alt="{\displaystyle i,j\neq 1,n}"></noscript><span class="lazy-image-placeholder" style="width: 9.484ex;height: 2.676ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875dc47ba5de13bbbafbfa58a3c9dbc029b1ede8" data-alt="{\displaystyle i,j\neq 1,n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. One can do this using any operation that one could normally perform without changing the value of the determinant, such as adding a multiple of one row to another.</li> <li>Create an (<i>n</i> − 1) × (<i>n</i> − 1) matrix B, consisting of the determinants of every 2 × 2 submatrix of A. Explicitly, we write <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i,j}={\begin{vmatrix}a_{i,j}&a_{i,j+1}\\a_{i+1,j}&a_{i+1,j+1}\end{vmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</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"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i,j}={\begin{vmatrix}a_{i,j}&a_{i,j+1}\\a_{i+1,j}&a_{i+1,j+1}\end{vmatrix}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350413dcbf48447a0c751933dbee1369a4218f57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.675ex; height:6.509ex;" alt="{\displaystyle b_{i,j}={\begin{vmatrix}a_{i,j}&a_{i,j+1}\\a_{i+1,j}&a_{i+1,j+1}\end{vmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 23.675ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350413dcbf48447a0c751933dbee1369a4218f57" data-alt="{\displaystyle b_{i,j}={\begin{vmatrix}a_{i,j}&a_{i,j+1}\\a_{i+1,j}&a_{i+1,j+1}\end{vmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></li> <li>Using this (<i>n</i> − 1) × (<i>n</i> − 1) matrix, perform step 2 to obtain an (<i>n</i> − 2) × (<i>n</i> − 2) matrix C. Divide each term in C by the corresponding term in the interior of A so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{i,j}={\begin{vmatrix}b_{i,j}&b_{i,j+1}\\b_{i+1,j}&b_{i+1,j+1}\end{vmatrix}}/a_{i+1,j+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i,j}={\begin{vmatrix}b_{i,j}&b_{i,j+1}\\b_{i+1,j}&b_{i+1,j+1}\end{vmatrix}}/a_{i+1,j+1}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f760d58d969863af7108c2a0d7faff3ec628f20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.101ex; height:6.509ex;" alt="{\displaystyle c_{i,j}={\begin{vmatrix}b_{i,j}&b_{i,j+1}\\b_{i+1,j}&b_{i+1,j+1}\end{vmatrix}}/a_{i+1,j+1}}"></noscript><span class="lazy-image-placeholder" style="width: 31.101ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f760d58d969863af7108c2a0d7faff3ec628f20" data-alt="{\displaystyle c_{i,j}={\begin{vmatrix}b_{i,j}&b_{i,j+1}\\b_{i+1,j}&b_{i+1,j+1}\end{vmatrix}}/a_{i+1,j+1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>.</li> <li>Let A = B, and B = C. Repeat step 3 as necessary until the 1 × 1 matrix is found; its only entry is the determinant.</li></ol> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Examples">Examples</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=2" title="Edit section: Examples" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-2 collapsible-block" id="mf-section-2"> <div class="mw-heading mw-heading3"><h3 id="Without_zeros">Without zeros</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=3" title="Edit section: Without zeros" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>One wishes to find </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{vmatrix}-2&-1&-1&-4\\-1&-2&-1&-6\\-1&-1&2&4\\2&1&-3&-8\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> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>8</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}-2&-1&-1&-4\\-1&-2&-1&-6\\-1&-1&2&4\\2&1&-3&-8\end{vmatrix}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89228a9922b1999aca4c05c53f17a61c632b20a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:21.542ex; height:12.509ex;" alt="{\displaystyle {\begin{vmatrix}-2&-1&-1&-4\\-1&-2&-1&-6\\-1&-1&2&4\\2&1&-3&-8\end{vmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 21.542ex;height: 12.509ex;vertical-align: -5.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89228a9922b1999aca4c05c53f17a61c632b20a1" data-alt="{\displaystyle {\begin{vmatrix}-2&-1&-1&-4\\-1&-2&-1&-6\\-1&-1&2&4\\2&1&-3&-8\end{vmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>All of the interior elements are non-zero, so there is no need to re-arrange the matrix. </p><p>We make a matrix of its 2 × 2 submatrices. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}-2&-1\\-1&-2\end{vmatrix}}&{\begin{vmatrix}-1&-1\\-2&-1\end{vmatrix}}&{\begin{vmatrix}-1&-4\\-1&-6\end{vmatrix}}\\\\{\begin{vmatrix}-1&-2\\-1&-1\end{vmatrix}}&{\begin{vmatrix}-2&-1\\-1&2\end{vmatrix}}&{\begin{vmatrix}-1&-6\\2&4\end{vmatrix}}\\\\{\begin{vmatrix}-1&-1\\2&1\end{vmatrix}}&{\begin{vmatrix}-1&2\\1&-3\end{vmatrix}}&{\begin{vmatrix}2&4\\-3&-8\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}3&-1&2\\-1&-5&8\\1&1&-4\end{bmatrix}}.}"> <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> <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> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> <mtd> <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> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>8</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </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> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\begin{vmatrix}-2&-1\\-1&-2\end{vmatrix}}&{\begin{vmatrix}-1&-1\\-2&-1\end{vmatrix}}&{\begin{vmatrix}-1&-4\\-1&-6\end{vmatrix}}\\\\{\begin{vmatrix}-1&-2\\-1&-1\end{vmatrix}}&{\begin{vmatrix}-2&-1\\-1&2\end{vmatrix}}&{\begin{vmatrix}-1&-6\\2&4\end{vmatrix}}\\\\{\begin{vmatrix}-1&-1\\2&1\end{vmatrix}}&{\begin{vmatrix}-1&2\\1&-3\end{vmatrix}}&{\begin{vmatrix}2&4\\-3&-8\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}3&-1&2\\-1&-5&8\\1&1&-4\end{bmatrix}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700f53f134fccbcaa4056455777277e72232e3ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:60.578ex; height:25.509ex;" alt="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}-2&-1\\-1&-2\end{vmatrix}}&{\begin{vmatrix}-1&-1\\-2&-1\end{vmatrix}}&{\begin{vmatrix}-1&-4\\-1&-6\end{vmatrix}}\\\\{\begin{vmatrix}-1&-2\\-1&-1\end{vmatrix}}&{\begin{vmatrix}-2&-1\\-1&2\end{vmatrix}}&{\begin{vmatrix}-1&-6\\2&4\end{vmatrix}}\\\\{\begin{vmatrix}-1&-1\\2&1\end{vmatrix}}&{\begin{vmatrix}-1&2\\1&-3\end{vmatrix}}&{\begin{vmatrix}2&4\\-3&-8\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}3&-1&2\\-1&-5&8\\1&1&-4\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 60.578ex;height: 25.509ex;vertical-align: -12.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/700f53f134fccbcaa4056455777277e72232e3ce" data-alt="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}-2&-1\\-1&-2\end{vmatrix}}&{\begin{vmatrix}-1&-1\\-2&-1\end{vmatrix}}&{\begin{vmatrix}-1&-4\\-1&-6\end{vmatrix}}\\\\{\begin{vmatrix}-1&-2\\-1&-1\end{vmatrix}}&{\begin{vmatrix}-2&-1\\-1&2\end{vmatrix}}&{\begin{vmatrix}-1&-6\\2&4\end{vmatrix}}\\\\{\begin{vmatrix}-1&-1\\2&1\end{vmatrix}}&{\begin{vmatrix}-1&2\\1&-3\end{vmatrix}}&{\begin{vmatrix}2&4\\-3&-8\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}3&-1&2\\-1&-5&8\\1&1&-4\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>We then find another matrix of determinants: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}3&-1\\-1&-5\end{vmatrix}}&{\begin{vmatrix}-1&2\\-5&8\end{vmatrix}}\\\\{\begin{vmatrix}-1&-5\\1&1\end{vmatrix}}&{\begin{vmatrix}-5&8\\1&-4\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}-16&2\\4&12\end{bmatrix}}.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>16</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>12</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\begin{vmatrix}3&-1\\-1&-5\end{vmatrix}}&{\begin{vmatrix}-1&2\\-5&8\end{vmatrix}}\\\\{\begin{vmatrix}-1&-5\\1&1\end{vmatrix}}&{\begin{vmatrix}-5&8\\1&-4\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}-16&2\\4&12\end{bmatrix}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456b84a56975c3245943fdb80da29fe8e3a3db73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:42.525ex; height:15.843ex;" alt="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}3&-1\\-1&-5\end{vmatrix}}&{\begin{vmatrix}-1&2\\-5&8\end{vmatrix}}\\\\{\begin{vmatrix}-1&-5\\1&1\end{vmatrix}}&{\begin{vmatrix}-5&8\\1&-4\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}-16&2\\4&12\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 42.525ex;height: 15.843ex;vertical-align: -7.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/456b84a56975c3245943fdb80da29fe8e3a3db73" data-alt="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}3&-1\\-1&-5\end{vmatrix}}&{\begin{vmatrix}-1&2\\-5&8\end{vmatrix}}\\\\{\begin{vmatrix}-1&-5\\1&1\end{vmatrix}}&{\begin{vmatrix}-5&8\\1&-4\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}-16&2\\4&12\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>We must then divide each element by the corresponding element of our original matrix. The interior of the original matrix is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}-2&-1\\-1&2\end{bmatrix}}}"> <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> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}-2&-1\\-1&2\end{bmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4accdb4eead15d4a482315be63d1424e98665dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.47ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}-2&-1\\-1&2\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 11.47ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4accdb4eead15d4a482315be63d1424e98665dc2" data-alt="{\displaystyle {\begin{bmatrix}-2&-1\\-1&2\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, so after dividing we get <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}8&-2\\-4&6\end{bmatrix}}}"> <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>8</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}8&-2\\-4&6\end{bmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9cd65c28f5af7eb46a19090992c8912c07bdb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.47ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}8&-2\\-4&6\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 11.47ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9cd65c28f5af7eb46a19090992c8912c07bdb0" data-alt="{\displaystyle {\begin{bmatrix}8&-2\\-4&6\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. The process must be repeated to arrive at a 1 × 1 matrix. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}8&-2\\-4&6\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}40\end{bmatrix}}.}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </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>40</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}{\begin{vmatrix}8&-2\\-4&6\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}40\end{bmatrix}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2294c145fc5fa4eba7865e5badf2dd16d8505a2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.631ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}8&-2\\-4&6\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}40\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 21.631ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2294c145fc5fa4eba7865e5badf2dd16d8505a2e" data-alt="{\displaystyle {\begin{bmatrix}{\begin{vmatrix}8&-2\\-4&6\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}40\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> Dividing by the interior of the 3 × 3 matrix, which is just −5, gives <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}-8\end{bmatrix}}}"> <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> <mo>−</mo> <mn>8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}-8\end{bmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab859a78a5dd94230faaa0f27b87530b8067dbb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.016ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}-8\end{bmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 5.016ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab859a78a5dd94230faaa0f27b87530b8067dbb5" data-alt="{\displaystyle {\begin{bmatrix}-8\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and −8 is indeed the determinant of the original matrix. </p> <div class="mw-heading mw-heading3"><h3 id="With_zeros">With zeros</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=4" title="Edit section: With zeros" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Simply writing out the matrices: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}2&-1&2&1&-3\\1&2&1&-1&2\\1&-1&-2&-1&-1\\2&1&-1&-2&-1\\1&-2&-1&-1&2\end{bmatrix}}\to {\begin{bmatrix}5&-5&-3&-1\\-3&-3&-3&3\\3&3&3&-1\\-5&-3&-1&-5\end{bmatrix}}\to {\begin{bmatrix}-15&6&12\\0&0&6\\6&-6&8\end{bmatrix}}.}"> <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>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>15</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>12</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}2&-1&2&1&-3\\1&2&1&-1&2\\1&-1&-2&-1&-1\\2&1&-1&-2&-1\\1&-2&-1&-1&2\end{bmatrix}}\to {\begin{bmatrix}5&-5&-3&-1\\-3&-3&-3&3\\3&3&3&-1\\-5&-3&-1&-5\end{bmatrix}}\to {\begin{bmatrix}-15&6&12\\0&0&6\\6&-6&8\end{bmatrix}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c392cd9b283efbe1eb55b3672149fbfc8de6c14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:74.69ex; height:15.843ex;" alt="{\displaystyle {\begin{bmatrix}2&-1&2&1&-3\\1&2&1&-1&2\\1&-1&-2&-1&-1\\2&1&-1&-2&-1\\1&-2&-1&-1&2\end{bmatrix}}\to {\begin{bmatrix}5&-5&-3&-1\\-3&-3&-3&3\\3&3&3&-1\\-5&-3&-1&-5\end{bmatrix}}\to {\begin{bmatrix}-15&6&12\\0&0&6\\6&-6&8\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 74.69ex;height: 15.843ex;vertical-align: -7.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c392cd9b283efbe1eb55b3672149fbfc8de6c14a" data-alt="{\displaystyle {\begin{bmatrix}2&-1&2&1&-3\\1&2&1&-1&2\\1&-1&-2&-1&-1\\2&1&-1&-2&-1\\1&-2&-1&-1&2\end{bmatrix}}\to {\begin{bmatrix}5&-5&-3&-1\\-3&-3&-3&3\\3&3&3&-1\\-5&-3&-1&-5\end{bmatrix}}\to {\begin{bmatrix}-15&6&12\\0&0&6\\6&-6&8\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Here we run into trouble. If we continue the process, we will eventually be dividing by 0. We can perform four row exchanges on the initial matrix to preserve the determinant and repeat the process, with most of the determinants precalculated: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&1&-1&2\\1&-1&-2&-1&-1\\2&1&-1&-2&-1\\1&-2&-1&-1&2\\2&-1&2&1&-3\end{bmatrix}}\to {\begin{bmatrix}-3&-3&-3&3\\3&3&3&-1\\-5&-3&-1&-5\\3&-5&1&1\end{bmatrix}}\to {\begin{bmatrix}0&0&6\\6&-6&8\\-17&8&-4\end{bmatrix}}\to {\begin{bmatrix}0&12\\18&40\end{bmatrix}}\to {\begin{bmatrix}36\end{bmatrix}}.}"> <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>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</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>3</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> <mtd> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>17</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>12</mn> </mtd> </mtr> <mtr> <mtd> <mn>18</mn> </mtd> <mtd> <mn>40</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>36</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&1&-1&2\\1&-1&-2&-1&-1\\2&1&-1&-2&-1\\1&-2&-1&-1&2\\2&-1&2&1&-3\end{bmatrix}}\to {\begin{bmatrix}-3&-3&-3&3\\3&3&3&-1\\-5&-3&-1&-5\\3&-5&1&1\end{bmatrix}}\to {\begin{bmatrix}0&0&6\\6&-6&8\\-17&8&-4\end{bmatrix}}\to {\begin{bmatrix}0&12\\18&40\end{bmatrix}}\to {\begin{bmatrix}36\end{bmatrix}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5a49c760403e441938e88003811a46b912d5d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:97.113ex; height:15.843ex;" alt="{\displaystyle {\begin{bmatrix}1&2&1&-1&2\\1&-1&-2&-1&-1\\2&1&-1&-2&-1\\1&-2&-1&-1&2\\2&-1&2&1&-3\end{bmatrix}}\to {\begin{bmatrix}-3&-3&-3&3\\3&3&3&-1\\-5&-3&-1&-5\\3&-5&1&1\end{bmatrix}}\to {\begin{bmatrix}0&0&6\\6&-6&8\\-17&8&-4\end{bmatrix}}\to {\begin{bmatrix}0&12\\18&40\end{bmatrix}}\to {\begin{bmatrix}36\end{bmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 97.113ex;height: 15.843ex;vertical-align: -7.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5a49c760403e441938e88003811a46b912d5d3" data-alt="{\displaystyle {\begin{bmatrix}1&2&1&-1&2\\1&-1&-2&-1&-1\\2&1&-1&-2&-1\\1&-2&-1&-1&2\\2&-1&2&1&-3\end{bmatrix}}\to {\begin{bmatrix}-3&-3&-3&3\\3&3&3&-1\\-5&-3&-1&-5\\3&-5&1&1\end{bmatrix}}\to {\begin{bmatrix}0&0&6\\6&-6&8\\-17&8&-4\end{bmatrix}}\to {\begin{bmatrix}0&12\\18&40\end{bmatrix}}\to {\begin{bmatrix}36\end{bmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Hence, we arrive at a determinant of 36. </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Desnanot–Jacobi_identity_and_proof_of_correctness_of_the_condensation_algorithm"><span id="Desnanot.E2.80.93Jacobi_identity_and_proof_of_correctness_of_the_condensation_algorithm"></span>Desnanot–Jacobi identity and proof of correctness of the condensation algorithm</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=5" title="Edit section: Desnanot–Jacobi identity and proof of correctness of the condensation algorithm" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-3 collapsible-block" id="mf-section-3"> <p>The proof that the condensation method computes the determinant of the matrix if no divisions by zero are encountered is based on an identity known as the <b>Desnanot–Jacobi identity</b> (1841) or, more generally, the <a href="/wiki/Sylvester%27s_determinant_identity" title="Sylvester's determinant identity">Sylvester determinant identity</a> (1851).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=(m_{i,j})_{i,j=1}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=(m_{i,j})_{i,j=1}^{k}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7369f17f450fe2f3f2b326479ec1f8cd6186aceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.36ex; height:3.509ex;" alt="{\displaystyle M=(m_{i,j})_{i,j=1}^{k}}"></noscript><span class="lazy-image-placeholder" style="width: 15.36ex;height: 3.509ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7369f17f450fe2f3f2b326479ec1f8cd6186aceb" data-alt="{\displaystyle M=(m_{i,j})_{i,j=1}^{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> be a square matrix, and for each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i,j\leq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≤</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i,j\leq k}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10faadbed46876b6037237a8e7ee613f0bd4039f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.365ex; height:2.509ex;" alt="{\displaystyle 1\leq i,j\leq k}"></noscript><span class="lazy-image-placeholder" style="width: 11.365ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10faadbed46876b6037237a8e7ee613f0bd4039f" data-alt="{\displaystyle 1\leq i,j\leq k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, denote by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}^{j}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44290a8104a12869461798f1e0e77956e86a385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.408ex; height:3.509ex;" alt="{\displaystyle M_{i}^{j}}"></noscript><span class="lazy-image-placeholder" style="width: 3.408ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44290a8104a12869461798f1e0e77956e86a385" data-alt="{\displaystyle M_{i}^{j}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> the matrix that results from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> by deleting the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 0.802ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" data-alt="{\displaystyle i}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th row and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 0.985ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" data-alt="{\displaystyle j}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th column. Similarly, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i,j,p,q\leq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>≤</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i,j,p,q\leq k}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/058c08eee45b42fc9a984fa30197dfec569d6676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.672ex; height:2.509ex;" alt="{\displaystyle 1\leq i,j,p,q\leq k}"></noscript><span class="lazy-image-placeholder" style="width: 15.672ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/058c08eee45b42fc9a984fa30197dfec569d6676" data-alt="{\displaystyle 1\leq i,j,p,q\leq k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, denote by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i,j}^{p,q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i,j}^{p,q}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b32bdc5404e862f186f19927f7aa91fa80395a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.772ex; height:3.509ex;" alt="{\displaystyle M_{i,j}^{p,q}}"></noscript><span class="lazy-image-placeholder" style="width: 4.772ex;height: 3.509ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b32bdc5404e862f186f19927f7aa91fa80395a" data-alt="{\displaystyle M_{i,j}^{p,q}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> the matrix that results from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> by deleting the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 0.802ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" data-alt="{\displaystyle i}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 0.985ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" data-alt="{\displaystyle j}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th rows and the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></noscript><span class="lazy-image-placeholder" style="width: 1.259ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" data-alt="{\displaystyle p}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></noscript><span class="lazy-image-placeholder" style="width: 1.07ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" data-alt="{\displaystyle q}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th columns. </p> <div class="mw-heading mw-heading3"><h3 id="Desnanot–Jacobi_identity"><span id="Desnanot.E2.80.93Jacobi_identity"></span>Desnanot–Jacobi identity</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=6" title="Edit section: Desnanot–Jacobi identity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(M)\det(M_{1,k}^{1,k})=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(M)\det(M_{1,k}^{1,k})=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1}).}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc9696aaa8a2a8df1adae1a101f9a229cfcca3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:59.57ex; height:3.843ex;" alt="{\displaystyle \det(M)\det(M_{1,k}^{1,k})=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1}).}"></noscript><span class="lazy-image-placeholder" style="width: 59.57ex;height: 3.843ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bc9696aaa8a2a8df1adae1a101f9a229cfcca3a" data-alt="{\displaystyle \det(M)\det(M_{1,k}^{1,k})=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1}).}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Proof_of_the_correctness_of_Dodgson_condensation">Proof of the correctness of Dodgson condensation</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=7" title="Edit section: Proof of the correctness of Dodgson condensation" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>Rewrite the identity as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(M)={\frac {\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1})}{\det(M_{1,k}^{1,k})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(M)={\frac {\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1})}{\det(M_{1,k}^{1,k})}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64931792d2c835e366a4628a2e53f7ae88870a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:50.113ex; height:7.843ex;" alt="{\displaystyle \det(M)={\frac {\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1})}{\det(M_{1,k}^{1,k})}}.}"></noscript><span class="lazy-image-placeholder" style="width: 50.113ex;height: 7.843ex;vertical-align: -3.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64931792d2c835e366a4628a2e53f7ae88870a36" data-alt="{\displaystyle \det(M)={\frac {\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1})}{\det(M_{1,k}^{1,k})}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p>Now note that by induction it follows that when applying the Dodgson condensation procedure to a square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, the matrix in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th stage of the computation (where the first stage <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></noscript><span class="lazy-image-placeholder" style="width: 5.472ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" data-alt="{\displaystyle k=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> corresponds to the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> itself) consists of all the <i>connected minors</i> of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, where a connected minor is the determinant of a connected <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\times k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times k}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bcf9346bcb189917b6b49c4331b4483f4a4a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.263ex; height:2.176ex;" alt="{\displaystyle k\times k}"></noscript><span class="lazy-image-placeholder" style="width: 5.263ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bcf9346bcb189917b6b49c4331b4483f4a4a2c" data-alt="{\displaystyle k\times k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> sub-block of adjacent entries of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. In particular, in the last stage <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b8cc6fcba0c0b24656b0fb33414d2e6cffb83c" 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}"></noscript><span class="lazy-image-placeholder" style="width: 5.704ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4b8cc6fcba0c0b24656b0fb33414d2e6cffb83c" data-alt="{\displaystyle k=n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, one gets a matrix containing a single element equal to the unique connected minor of order <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.395ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" data-alt="{\displaystyle n}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, namely the determinant of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Proof_of_the_Desnanot-Jacobi_identity">Proof of the Desnanot-Jacobi identity</h3><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=8" title="Edit section: Proof of the Desnanot-Jacobi identity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div> <p>We follow the treatment in the book <i>Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture</i>;<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> an alternative combinatorial proof was given in a paper by <a href="/wiki/Doron_Zeilberger" title="Doron Zeilberger">Doron Zeilberger</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p><br> </p><p><br> Denote <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,j}=(-1)^{i+j}\det(M_{i}^{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <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> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}=(-1)^{i+j}\det(M_{i}^{j})}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679509e0dd4b5f5cb3166d55e9919e82de1457d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.633ex; height:3.509ex;" alt="{\displaystyle a_{i,j}=(-1)^{i+j}\det(M_{i}^{j})}"></noscript><span class="lazy-image-placeholder" style="width: 22.633ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/679509e0dd4b5f5cb3166d55e9919e82de1457d8" data-alt="{\displaystyle a_{i,j}=(-1)^{i+j}\det(M_{i}^{j})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (up to sign, the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></noscript><span class="lazy-image-placeholder" style="width: 4.604ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" data-alt="{\displaystyle (i,j)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th minor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>), and define a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\times k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times k}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bcf9346bcb189917b6b49c4331b4483f4a4a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.263ex; height:2.176ex;" alt="{\displaystyle k\times k}"></noscript><span class="lazy-image-placeholder" style="width: 5.263ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bcf9346bcb189917b6b49c4331b4483f4a4a2c" data-alt="{\displaystyle k\times k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M'}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3b2ef3304c46b5e7859eec0b1bc057c8eb3f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.183ex; height:2.509ex;" alt="{\displaystyle M'}"></noscript><span class="lazy-image-placeholder" style="width: 3.183ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3b2ef3304c46b5e7859eec0b1bc057c8eb3f75" data-alt="{\displaystyle M'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> by <br> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M'={\begin{pmatrix}a_{1,1}&0&0&0&\ldots &0&a_{k,1}\\a_{1,2}&1&0&0&\ldots &0&a_{k,2}\\a_{1,3}&0&1&0&\ldots &0&a_{k,3}\\a_{1,4}&0&0&1&\ldots &0&a_{k,4}\\\vdots &\vdots &\vdots &\vdots &&\vdots &\vdots \\a_{1,k-1}&0&0&0&\ldots &1&a_{k,k-1}\\a_{1,k}&0&0&0&\ldots &0&a_{k,k}\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mo>′</mo> </msup> <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> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>4</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M'={\begin{pmatrix}a_{1,1}&0&0&0&\ldots &0&a_{k,1}\\a_{1,2}&1&0&0&\ldots &0&a_{k,2}\\a_{1,3}&0&1&0&\ldots &0&a_{k,3}\\a_{1,4}&0&0&1&\ldots &0&a_{k,4}\\\vdots &\vdots &\vdots &\vdots &&\vdots &\vdots \\a_{1,k-1}&0&0&0&\ldots &1&a_{k,k-1}\\a_{1,k}&0&0&0&\ldots &0&a_{k,k}\end{pmatrix}}.}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5d45e4f80a21bb83de157e3fa7f4a5765b2170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.838ex; width:44.487ex; height:24.843ex;" alt="{\displaystyle M'={\begin{pmatrix}a_{1,1}&0&0&0&\ldots &0&a_{k,1}\\a_{1,2}&1&0&0&\ldots &0&a_{k,2}\\a_{1,3}&0&1&0&\ldots &0&a_{k,3}\\a_{1,4}&0&0&1&\ldots &0&a_{k,4}\\\vdots &\vdots &\vdots &\vdots &&\vdots &\vdots \\a_{1,k-1}&0&0&0&\ldots &1&a_{k,k-1}\\a_{1,k}&0&0&0&\ldots &0&a_{k,k}\end{pmatrix}}.}"></noscript><span class="lazy-image-placeholder" style="width: 44.487ex;height: 24.843ex;vertical-align: -11.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b5d45e4f80a21bb83de157e3fa7f4a5765b2170" data-alt="{\displaystyle M'={\begin{pmatrix}a_{1,1}&0&0&0&\ldots &0&a_{k,1}\\a_{1,2}&1&0&0&\ldots &0&a_{k,2}\\a_{1,3}&0&1&0&\ldots &0&a_{k,3}\\a_{1,4}&0&0&1&\ldots &0&a_{k,4}\\\vdots &\vdots &\vdots &\vdots &&\vdots &\vdots \\a_{1,k-1}&0&0&0&\ldots &1&a_{k,k-1}\\a_{1,k}&0&0&0&\ldots &0&a_{k,k}\end{pmatrix}}.}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p><br> (Note that the first and last column of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M'}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3b2ef3304c46b5e7859eec0b1bc057c8eb3f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.183ex; height:2.509ex;" alt="{\displaystyle M'}"></noscript><span class="lazy-image-placeholder" style="width: 3.183ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3b2ef3304c46b5e7859eec0b1bc057c8eb3f75" data-alt="{\displaystyle M'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> are equal to those of the <a href="/wiki/Adjugate_matrix" title="Adjugate matrix">adjugate matrix</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><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}"></noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). The identity is now obtained by computing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(MM')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(MM')}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1889cb6db9b2f8cfabc08118e4cb4525d94970c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.665ex; height:3.009ex;" alt="{\displaystyle \det(MM')}"></noscript><span class="lazy-image-placeholder" style="width: 10.665ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1889cb6db9b2f8cfabc08118e4cb4525d94970c" data-alt="{\displaystyle \det(MM')}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> in two ways. First, we can directly compute the matrix product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle MM'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msup> <mi>M</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MM'}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eddde549ebf6958addcc28ba12739e0243cef53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.626ex; height:2.509ex;" alt="{\displaystyle MM'}"></noscript><span class="lazy-image-placeholder" style="width: 5.626ex;height: 2.509ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eddde549ebf6958addcc28ba12739e0243cef53" data-alt="{\displaystyle MM'}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (using simple properties of the adjugate matrix, or alternatively using the formula for the expansion of a matrix determinant in terms of a row or a column) to arrive at <br> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle MM'={\begin{pmatrix}\det(M)&m_{1,2}&m_{1,3}&\ldots &m_{1,k-1}&0\\0&m_{2,2}&m_{2,3}&\ldots &m_{2,k-1}&0\\0&m_{3,2}&m_{3,3}&\ldots &m_{3,k-1}&0\\\vdots &\vdots &\vdots &&\vdots &\vdots &\vdots \\0&m_{k-1,2}&m_{k-1,3}&\ldots &m_{k-1,k-1}&0\\0&m_{k,2}&m_{k,3}&\ldots &m_{k,k-1}&\det(M)\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd></mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MM'={\begin{pmatrix}\det(M)&m_{1,2}&m_{1,3}&\ldots &m_{1,k-1}&0\\0&m_{2,2}&m_{2,3}&\ldots &m_{2,k-1}&0\\0&m_{3,2}&m_{3,3}&\ldots &m_{3,k-1}&0\\\vdots &\vdots &\vdots &&\vdots &\vdots &\vdots \\0&m_{k-1,2}&m_{k-1,3}&\ldots &m_{k-1,k-1}&0\\0&m_{k,2}&m_{k,3}&\ldots &m_{k,k-1}&\det(M)\end{pmatrix}}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e65d33ac2d405da4e246daff53f89bb12d366c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.005ex; width:67.472ex; height:21.176ex;" alt="{\displaystyle MM'={\begin{pmatrix}\det(M)&m_{1,2}&m_{1,3}&\ldots &m_{1,k-1}&0\\0&m_{2,2}&m_{2,3}&\ldots &m_{2,k-1}&0\\0&m_{3,2}&m_{3,3}&\ldots &m_{3,k-1}&0\\\vdots &\vdots &\vdots &&\vdots &\vdots &\vdots \\0&m_{k-1,2}&m_{k-1,3}&\ldots &m_{k-1,k-1}&0\\0&m_{k,2}&m_{k,3}&\ldots &m_{k,k-1}&\det(M)\end{pmatrix}}}"></noscript><span class="lazy-image-placeholder" style="width: 67.472ex;height: 21.176ex;vertical-align: -10.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e65d33ac2d405da4e246daff53f89bb12d366c" data-alt="{\displaystyle MM'={\begin{pmatrix}\det(M)&m_{1,2}&m_{1,3}&\ldots &m_{1,k-1}&0\\0&m_{2,2}&m_{2,3}&\ldots &m_{2,k-1}&0\\0&m_{3,2}&m_{3,3}&\ldots &m_{3,k-1}&0\\\vdots &\vdots &\vdots &&\vdots &\vdots &\vdots \\0&m_{k-1,2}&m_{k-1,3}&\ldots &m_{k-1,k-1}&0\\0&m_{k,2}&m_{k,3}&\ldots &m_{k,k-1}&\det(M)\end{pmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span></dd></dl> <p><br> where we use <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5b8059baa5be84593d0400fa3a5b133286494c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.975ex; height:2.343ex;" alt="{\displaystyle m_{i,j}}"></noscript><span class="lazy-image-placeholder" style="width: 3.975ex;height: 2.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5b8059baa5be84593d0400fa3a5b133286494c" data-alt="{\displaystyle m_{i,j}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> to denote the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></noscript><span class="lazy-image-placeholder" style="width: 4.604ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" data-alt="{\displaystyle (i,j)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>-th entry of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></noscript><span class="lazy-image-placeholder" style="width: 2.442ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" data-alt="{\displaystyle M}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. The determinant of this matrix is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(M)^{2}\cdot \det(M_{1,k}^{1,k})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(M)^{2}\cdot \det(M_{1,k}^{1,k})}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9e8722db77e144014120049b58e5e6889c2c58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.12ex; height:3.843ex;" alt="{\displaystyle \det(M)^{2}\cdot \det(M_{1,k}^{1,k})}"></noscript><span class="lazy-image-placeholder" style="width: 20.12ex;height: 3.843ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9e8722db77e144014120049b58e5e6889c2c58" data-alt="{\displaystyle \det(M)^{2}\cdot \det(M_{1,k}^{1,k})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. <br>Second, this is equal to the product of the determinants, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(M)\cdot \det(M')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(M)\cdot \det(M')}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1f7d821941e581d0fb116f4409a61874ba6345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.382ex; height:3.009ex;" alt="{\displaystyle \det(M)\cdot \det(M')}"></noscript><span class="lazy-image-placeholder" style="width: 17.382ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1f7d821941e581d0fb116f4409a61874ba6345" data-alt="{\displaystyle \det(M)\cdot \det(M')}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>. But clearly <br> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(M')=a_{1,1}a_{k,k}-a_{k,1}a_{1,k}=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(M')=a_{1,1}a_{k,k}-a_{k,1}a_{1,k}=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1}),}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b587020afee50fcc44319d15170b8d50748002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:70.349ex; height:3.176ex;" alt="{\displaystyle \det(M')=a_{1,1}a_{k,k}-a_{k,1}a_{1,k}=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1}),}"></noscript><span class="lazy-image-placeholder" style="width: 70.349ex;height: 3.176ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b587020afee50fcc44319d15170b8d50748002" data-alt="{\displaystyle \det(M')=a_{1,1}a_{k,k}-a_{k,1}a_{1,k}=\det(M_{1}^{1})\det(M_{k}^{k})-\det(M_{1}^{k})\det(M_{k}^{1}),}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> <br> so the identity follows from equating the two expressions we obtained for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(MM')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(MM')}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1889cb6db9b2f8cfabc08118e4cb4525d94970c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.665ex; height:3.009ex;" alt="{\displaystyle \det(MM')}"></noscript><span class="lazy-image-placeholder" style="width: 10.665ex;height: 3.009ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1889cb6db9b2f8cfabc08118e4cb4525d94970c" data-alt="{\displaystyle \det(MM')}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> and dividing out by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(M)}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b3a4162384b57d6b54a8f253a28e3d874c8750" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.481ex; height:2.843ex;" alt="{\displaystyle \det(M)}"></noscript><span class="lazy-image-placeholder" style="width: 7.481ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b3a4162384b57d6b54a8f253a28e3d874c8750" data-alt="{\displaystyle \det(M)}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> (this is allowed if one thinks of the identities as polynomial identities over the ring of polynomials in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k^{2}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6423cd00e3559de92c4bc497066ff1b12bbfc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.265ex; height:2.676ex;" alt="{\displaystyle k^{2}}"></noscript><span class="lazy-image-placeholder" style="width: 2.265ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af6423cd00e3559de92c4bc497066ff1b12bbfc3" data-alt="{\displaystyle k^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span> indeterminate variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m_{i,j})_{i,j=1}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m_{i,j})_{i,j=1}^{k}}</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a04f5d68e0519434d29f0c869b02adb82fa1f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.819ex; height:3.509ex;" alt="{\displaystyle (m_{i,j})_{i,j=1}^{k}}"></noscript><span class="lazy-image-placeholder" style="width: 9.819ex;height: 3.509ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a04f5d68e0519434d29f0c869b02adb82fa1f3" data-alt="{\displaystyle (m_{i,j})_{i,j=1}^{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>). </p> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="References">References</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=9" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-4 collapsible-block" id="mf-section-4"> <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 class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFDodgson,_C._L.1866–1867" class="citation journal cs1">Dodgson, C. 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"On the relation between the minor determinants of linearly equivalent quadratic functions". <i>Philosophical Magazine</i>. <b>1</b>: 295–305.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+the+relation+between+the+minor+determinants+of+linearly+equivalent+quadratic+functions&rft.volume=1&rft.pages=295-305&rft.date=1851&rft.aulast=Sylvester&rft.aufirst=James+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADodgson+condensation" class="Z3988"></span> <br> Cited in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAkritasAkritasMalaschonok1996" class="citation journal cs1">Akritas, A. G.; Akritas, E. K.; Malaschonok, G. I. (1996). "Various proofs of Sylvester's (determinant) identity". <i>Mathematics and Computers in Simulation</i>. <b>42</b> (4–6): 585. <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%2FS0378-4754%2896%2900035-3">10.1016/S0378-4754(96)00035-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+and+Computers+in+Simulation&rft.atitle=Various+proofs+of+Sylvester%27s+%28determinant%29+identity&rft.volume=42&rft.issue=4%E2%80%936&rft.pages=585&rft.date=1996&rft_id=info%3Adoi%2F10.1016%2FS0378-4754%2896%2900035-3&rft.aulast=Akritas&rft.aufirst=A.+G.&rft.au=Akritas%2C+E.+K.&rft.au=Malaschonok%2C+G.+I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADodgson+condensation" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBressoud1999" class="citation book cs1">Bressoud, David (1999). <i>Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781316582756" title="Special:BookSources/9781316582756"><bdi>9781316582756</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proofs+and+Confirmations%3A+The+Story+of+the+Alternating+Sign+Matrix+Conjecture&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=9781316582756&rft.aulast=Bressoud&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADodgson+condensation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZeilberger1997" class="citation journal cs1">Zeilberger, Doron (1997). <a rel="nofollow" class="external text" href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v4i2r22">"Dodgson's Determinant-Evaluation Rule Proved by Two-Timing Men and Women"</a>. <i>Electron. J. Comb</i>. <b>4</b> (2): article R22. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.37236%2F1337">10.37236/1337</a></span><span class="reference-accessdate">. Retrieved <span class="nowrap">October 27,</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electron.+J.+Comb.&rft.atitle=Dodgson%27s+Determinant-Evaluation+Rule+Proved+by+Two-Timing+Men+and+Women&rft.volume=4&rft.issue=2&rft.pages=article+R22&rft.date=1997&rft_id=info%3Adoi%2F10.37236%2F1337&rft.aulast=Zeilberger&rft.aufirst=Doron&rft_id=https%3A%2F%2Fwww.combinatorics.org%2Fojs%2Findex.php%2Feljc%2Farticle%2Fview%2Fv4i2r22&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADodgson+condensation" class="Z3988"></span></span> </li> </ol></div></div> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=10" title="Edit section: Further reading" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-5 collapsible-block" id="mf-section-5"> <ul><li><a href="/wiki/David_Bressoud" title="David Bressoud">Bressoud, David M.</a> and Propp, James, <a rel="nofollow" class="external text" href="https://www.ams.org/notices/199906/fea-bressoud.pdf">How the alternating sign matrix conjecture was solved</a>, <i>Notices of the American Mathematical Society</i>, 46 (1999), 637-646.</li> <li><a href="/wiki/D._Knuth" class="mw-redirect" title="D. Knuth">Knuth, Donald</a>, <a rel="nofollow" class="external text" href="http://www.emis.de/journals/EJC/Volume_3/PDFFiles/v3i2r5.pdf">Overlapping Pfaffians</a>, <i>Electronic Journal of Combinatorics</i>, <b>3</b> no. 2 (1996).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLotkin,_Mark1959" class="citation journal cs1">Lotkin, Mark (1959). "Note on the Method of Contractants". <i>The American Mathematical Monthly</i>. <b>66</b> (6): 476–479. <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%2F2310629">10.2307/2310629</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/2310629">2310629</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Note+on+the+Method+of+Contractants&rft.volume=66&rft.issue=6&rft.pages=476-479&rft.date=1959&rft_id=info%3Adoi%2F10.2307%2F2310629&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2310629%23id-name%3DJSTOR&rft.au=Lotkin%2C+Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADodgson+condensation" class="Z3988"></span></li> <li>Mills, William H., Robbins, David P., and Rumsey, Howard, Jr., Proof of the Macdonald conjecture, <i>Inventiones Mathematicae</i>, 66 (1982), 73-87.</li> <li>Mills, William H., Robbins, David P., and Rumsey, Howard, Jr., Alternating sign matrices and descending plane partitions, <i><a href="/wiki/Journal_of_Combinatorial_Theory" title="Journal of Combinatorial Theory">Journal of Combinatorial Theory</a>, Series A</i>, 34 (1983), 340-359.</li> <li>Robbins, David P., The story of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,7,42,429,7436,\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>42</mn> <mo>,</mo> <mn>429</mn> <mo>,</mo> <mn>7436</mn> <mo>,</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,7,42,429,7436,\cdots }</annotation> </semantics> </math></span><noscript><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7856bfe6d03a9249fcf6b2a8c010b9d325973cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.876ex; height:2.509ex;" alt="{\displaystyle 1,2,7,42,429,7436,\cdots }"></noscript><span class="lazy-image-placeholder" style="width: 22.876ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7856bfe6d03a9249fcf6b2a8c010b9d325973cc" data-alt="{\displaystyle 1,2,7,42,429,7436,\cdots }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> </span></span>, <i>The Mathematical Intelligencer</i>, 13 (1991), 12-19.</li></ul> </section><div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"><span class="indicator mf-icon mf-icon-expand mf-icon--small"></span><h2 id="External_links">External links</h2><span class="mw-editsection"> <a role="button" href="/w/index.php?title=Dodgson_condensation&action=edit&section=11" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> <span class="minerva-icon minerva-icon--edit"></span> <span>edit</span> </a> </span> </div><section class="mf-section-6 collapsible-block" id="mf-section-6"> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Dodgson_condensation"><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. 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Dodgson condensation - Wikipedia