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Disponibil în 17 limbi" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-17" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">17 limbi</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Jedni%C4%8Dkov%C3%A1_matice" title="Jedničková matice – cehă" lang="cs" hreflang="cs" data-title="Jedničková matice" data-language-autonym="Čeština" data-language-local-name="cehă" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%C4%95%D1%80%D1%80%D0%B5%D1%81%D0%B5%D0%BD_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B8" title="Пĕрресен матрици – ciuvașă" lang="cv" hreflang="cv" data-title="Пĕрресен матрици" data-language-autonym="Чӑвашла" data-language-local-name="ciuvașă" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Einsmatrix" title="Einsmatrix – germană" lang="de" hreflang="de" data-title="Einsmatrix" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82_%CE%B1%CF%80%CF%8C_%CE%BC%CE%BF%CE%BD%CE%AC%CE%B4%CE%B5%CF%82" title="Πίνακας από μονάδες – greacă" lang="el" hreflang="el" data-title="Πίνακας από μονάδες" data-language-autonym="Ελληνικά" data-language-local-name="greacă" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Matrix_of_ones" title="Matrix of ones – engleză" lang="en" hreflang="en" data-title="Matrix of ones" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%DB%8C%DA%A9%E2%80%8C%D9%87%D8%A7" title="ماتریس یک‌ها – persană" lang="fa" hreflang="fa" data-title="ماتریس یک‌ها" data-language-autonym="فارسی" data-language-local-name="persană" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ykk%C3%B6smatriisi" title="Ykkösmatriisi – finlandeză" lang="fi" hreflang="fi" data-title="Ykkösmatriisi" data-language-autonym="Suomi" data-language-local-name="finlandeză" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matriz_de_uns" title="Matriz de uns – galiciană" lang="gl" hreflang="gl" data-title="Matriz de uns" data-language-autonym="Galego" data-language-local-name="galiciană" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%AA_%D7%90%D7%97%D7%93%D7%95%D7%AA" title="מטריצת אחדות – ebraică" lang="he" hreflang="he" data-title="מטריצת אחדות" data-language-autonym="עברית" data-language-local-name="ebraică" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_%D0%B5%D0%B4%D0%B8%D0%BD%D0%B8%D1%86" title="Матрица единиц – rusă" lang="ru" hreflang="ru" data-title="Матрица единиц" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matrika_enic" title="Matrika enic – slovenă" lang="sl" hreflang="sl" data-title="Matrika enic" data-language-autonym="Slovenščina" data-language-local-name="slovenă" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matrica_e_nj%C3%ABsheve" title="Matrica e njësheve – albaneză" lang="sq" hreflang="sq" data-title="Matrica e njësheve" data-language-autonym="Shqip" data-language-local-name="albaneză" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%92%E0%AE%A9%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%95%E0%AE%B3%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="ஒன்றுகளின் அணி – tamilă" lang="ta" hreflang="ta" data-title="ஒன்றுகளின் அணி" data-language-autonym="தமிழ்" data-language-local-name="tamilă" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A1%E0%B8%97%E0%B8%A3%E0%B8%B4%E0%B8%81%E0%B8%8B%E0%B9%8C%E0%B8%AB%E0%B8%99%E0%B8%B6%E0%B9%88%E0%B8%87" title="เมทริกซ์หนึ่ง – thailandeză" lang="th" hreflang="th" data-title="เมทริกซ์หนึ่ง" data-language-autonym="ไทย" data-language-local-name="thailandeză" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Birler_matrisi" title="Birler matrisi – turcă" lang="tr" hreflang="tr" data-title="Birler matrisi" data-language-autonym="Türkçe" data-language-local-name="turcă" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F_%D0%BE%D0%B4%D0%B8%D0%BD%D0%B8%D1%86%D1%8C" title="Матриця одиниць – ucraineană" lang="uk" hreflang="uk" data-title="Матриця одиниць" data-language-autonym="Українська" data-language-local-name="ucraineană" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%80%E7%9F%A9%E9%99%A3" title="一矩陣 – chineză" lang="zh" hreflang="zh" data-title="一矩陣" data-language-autonym="中文" data-language-local-name="chineză" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Aspect</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">mută în bara laterală</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ascunde</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De la Wikipedia, enciclopedia liberă</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ro" dir="ltr"><p>În <a href="/wiki/Algebr%C4%83_liniar%C4%83" title="Algebră liniară">algebra liniară</a>, o <b>matrice cu toate elementele 1</b><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> este o <a href="/wiki/Matrice" title="Matrice">matrice</a> în care fiecare element are <a href="/wiki/Valoare_(matematic%C4%83)" title="Valoare (matematică)">valoarea</a> <a href="/wiki/1_(cifr%C4%83)" title="1 (cifră)">1</a>.<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> Exemple de astfel de matrici: </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 J_{2}={\begin{pmatrix}1&1\\1&1\end{pmatrix}};\quad J_{3}={\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}};\quad J_{2,5}={\begin{pmatrix}1&1&1&1&1\\1&1&1&1&1\end{pmatrix}};\quad J_{1,2}={\begin{pmatrix}1&1\end{pmatrix}}.\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <mspace width="1em" /> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <mspace width="1em" /> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> <mspace width="1em" /> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{2}={\begin{pmatrix}1&1\\1&1\end{pmatrix}};\quad J_{3}={\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}};\quad J_{2,5}={\begin{pmatrix}1&1&1&1&1\\1&1&1&1&1\end{pmatrix}};\quad J_{1,2}={\begin{pmatrix}1&1\end{pmatrix}}.\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3bdab17eb6a11d3d2344861285f5f15ec919608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:85.624ex; height:9.176ex;" alt="{\displaystyle J_{2}={\begin{pmatrix}1&1\\1&1\end{pmatrix}};\quad J_{3}={\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}};\quad J_{2,5}={\begin{pmatrix}1&1&1&1&1\\1&1&1&1&1\end{pmatrix}};\quad J_{1,2}={\begin{pmatrix}1&1\end{pmatrix}}.\quad }"></span></dd></dl> <p>Unele surse numesc aceste matrici „<a href="/wiki/Matrice_unitate" title="Matrice unitate">matrice unitate</a>”,<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>, dar acest termen este folosit de obicei pentru matrici de alt tip. </p><p>Un <b>vector cu toate elementele 1</b> este o matrice cu toate elementele 1, <a href="/wiki/Vector_linie_%C8%99i_vector_coloan%C4%83" title="Vector linie și vector coloană">având o singură linie sau o singură coloană</a>. Ei nu trebuie confundați cu <i><a href="/wiki/Versor" title="Versor">versorii</a></i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proprietăți"><span id="Propriet.C4.83.C8.9Bi"></span>Proprietăți</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice_cu_toate_elementele_1&veaction=edit&section=1" title="Modifică secțiunea: Proprietăți" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice_cu_toate_elementele_1&action=edit&section=1" title="Edit section's source code: Proprietăți"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O matrice <span class="texhtml mvar" style="font-style:italic;">J</span> de dimensiuni <span class="texhtml mvar" style="font-style:italic;">n × n</span> cu toate elementele 1 are următoarele proprietăți: </p> <ul><li><a href="/wiki/Urm%C4%83_(algebr%C4%83)" title="Urmă (algebră)">Urma</a> lui <span class="texhtml mvar" style="font-style:italic;">J</span> este egală cu <span class="texhtml mvar" style="font-style:italic;">n</span>,<sup id="cite_ref-S_4-0" class="reference"><a href="#cite_note-S-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> și <a href="/wiki/Determinant_(matematic%C4%83)" title="Determinant (matematică)">determinantul</a> este 0 pentru <i>n</i> ≥ 2, dar 1 pentru <span class="texhtml mvar" style="font-style:italic;">n</span> = 1. (Se poate lua în considerare și cazul <span class="texhtml mvar" style="font-style:italic;">n</span> = 0, caz în care este vorba de o matrice vidă, al cărei determinant este 1.)</li> <li><a href="/w/index.php?title=Polinom_caracteristic&action=edit&redlink=1" class="new" title="Polinom caracteristic — pagină inexistentă">Polinomul caracteristic</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q849705" class="extiw" title="d:Q849705"><span title="polinom caracteristic la Wikidata">d</span></a>)</small></sup> al <span class="texhtml mvar" style="font-style:italic;">J</span> este <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 (x-n)x^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>n</mi> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-n)x^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6599421f1ca21b288b93849bdb4d2ce5f0655704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.023ex; height:3.176ex;" alt="{\displaystyle (x-n)x^{n-1}}"></span>.</li> <li><a href="/w/index.php?title=Polinom_minimal_(matrice)&action=edit&redlink=1" class="new" title="Polinom minimal (matrice) — pagină inexistentă">Polinomul minimal</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q1163608" class="extiw" title="d:Q1163608"><span title="Polinom minimal (matrice) la Wikidata">d</span></a>)</small></sup> al <span class="texhtml mvar" style="font-style:italic;">J</span> este <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 x^{2}-nx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>n</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-nx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4514288a32ab102be9de01ab8a5d9e25571f1f2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.949ex; height:2.843ex;" alt="{\displaystyle x^{2}-nx}"></span>.</li> <li><a href="/wiki/Rang_(algebr%C4%83_liniar%C4%83)#Rangul_unei_matrice" title="Rang (algebră liniară)">Rangul matricei</a> <span class="texhtml mvar" style="font-style:italic;">J</span> este 1, iar <a href="/wiki/Vectori_%C8%99i_valori_proprii" title="Vectori și valori proprii">vectorii proprii</a> sunt <span class="texhtml mvar" style="font-style:italic;">n</span> (cu <a href="/wiki/Multiplicitate" title="Multiplicitate">multiplicitatea</a> 1) și 0 (cu multiplicitatea <span class="texhtml mvar" style="font-style:italic;">n</span> − 1).<sup id="cite_ref-S_4-1" class="reference"><a href="#cite_note-S-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li><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^{k}=n^{k-1}J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{k}=n^{k-1}J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e82137d1a68be7482ce8ec1db7d4c5d7f04ee912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.768ex; height:2.676ex;" alt="{\displaystyle J^{k}=n^{k-1}J}"></span> pentru <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=1,2,\ldots \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1,2,\ldots \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8573517b9d5e12b1a88e9d529c82ae2764d82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.847ex; height:2.509ex;" alt="{\displaystyle k=1,2,\ldots \,.}"></span><sup id="cite_ref-timm_6-0" class="reference"><a href="#cite_note-timm-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><span class="texhtml mvar" style="font-style:italic;">J</span> este <i><a href="/wiki/Element_neutru" title="Element neutru">elementul neutru</a></i> pentru <a href="/wiki/Produs_Hadamard" title="Produs Hadamard">produsul Hadamard</a>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <p>Dacă <span class="texhtml mvar" style="font-style:italic;">J</span> este o matrice ale cărei elemente sunt <a href="/wiki/Num%C4%83r_real" title="Număr real">numere reale</a>, acestea au și următoarele proprietăți: </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">J</span> este o <a href="/w/index.php?title=Matrice_definit%C4%83&action=edit&redlink=1" class="new" title="Matrice definită — pagină inexistentă">matrice pozitivă semidefinită</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q77601250" class="extiw" title="d:Q77601250"><span title="matrice definită la Wikidata">d</span></a>)</small></sup>.</li> <li>Matricea <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 {\tfrac {1}{n}}J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mstyle> </mrow> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{n}}J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8508a124ee834914c8639399349b163f6fc496b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.294ex; height:3.343ex;" alt="{\displaystyle {\tfrac {1}{n}}J}"></span> este <a href="/wiki/Matrice_idempotent%C4%83" title="Matrice idempotentă">idempotentă</a>.<sup id="cite_ref-timm_6-1" class="reference"><a href="#cite_note-timm-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li><a href="/w/index.php?title=Exponen%C8%9Biala_unei_matrice&action=edit&redlink=1" class="new" title="Exponențiala unei matrice — pagină inexistentă">Exponențiala</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q1191722" class="extiw" title="d:Q1191722"><span title="exponențiala unei matrice la Wikidata">d</span></a>)</small></sup> lui <span class="texhtml mvar" style="font-style:italic;">J</span> este <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 \exp(J)=I+{\frac {e^{n}-1}{n}}J.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>J</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mi>n</mi> </mfrac> </mrow> <mi>J</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(J)=I+{\frac {e^{n}-1}{n}}J.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8879563edfd7ffcb555d0f65a76f6e91a3058c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.203ex; height:5.176ex;" alt="{\displaystyle \exp(J)=I+{\frac {e^{n}-1}{n}}J.}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Aplicații"><span id="Aplica.C8.9Bii"></span>Aplicații</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice_cu_toate_elementele_1&veaction=edit&section=2" title="Modifică secțiunea: Aplicații" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice_cu_toate_elementele_1&action=edit&section=2" title="Edit section's source code: Aplicații"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Matricea cu toate elementele 1 apare des în domeniul matematic al <a href="/wiki/Combinatoric%C4%83" title="Combinatorică">combinatoricii</a>, în special prin aplicarea metodelor algebrice la <a href="/wiki/Teoria_grafurilor" title="Teoria grafurilor">teoria grafurilor</a>. De exemplu, dacă <i>A</i> este <a href="/wiki/Matrice_de_adiacen%C8%9B%C4%83" title="Matrice de adiacență">matricea de adiacență</a> a unui <a href="/wiki/Graf" title="Graf">graf</a> neorientat <span class="texhtml mvar" style="font-style:italic;">G</span> cu <span class="texhtml mvar" style="font-style:italic;">n</span> noduri, iar <span class="texhtml mvar" style="font-style:italic;">J</span> este matricea cu toate elementele 1 de aceeași dimensiune, atunci <span class="texhtml mvar" style="font-style:italic;">G</span> este un <a href="/wiki/Graf_regulat" title="Graf regulat">graf regulat</a> <a href="/wiki/Dac%C4%83_%C8%99i_numai_dac%C4%83" title="Dacă și numai dacă">dacă și numai dacă</a> <span class="texhtml mvar" style="font-style:italic;">AJ</span> = <span class="texhtml mvar" style="font-style:italic;">JA</span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Un alt exemplu este că matricea apare în unele demonstrații algebrice ale <a href="/wiki/Formula_lui_Cayley" title="Formula lui Cayley">formulei lui Cayley</a>, care oferă numărul <a href="/w/index.php?title=Arbore_de_acoperire&action=edit&redlink=1" class="new" title="Arbore de acoperire — pagină inexistentă">arborilor de acoperire</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q831672" class="extiw" title="d:Q831672"><span title="arbore de acoperire la Wikidata">d</span></a>)</small></sup> ai unui <a href="/wiki/Graf_complet" title="Graf complet">graf complet</a>, folosind <a href="/w/index.php?title=Teorema_lui_Kirchhoff&action=edit&redlink=1" class="new" title="Teorema lui Kirchhoff — pagină inexistentă">teorema lui Kirchhoff</a><sup><small>⁠(<a href="https://www.wikidata.org/wiki/Q2226691" class="extiw" title="d:Q2226691"><span title="teorema lui Kirchhoff la Wikidata">d</span></a>)</small></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice_cu_toate_elementele_1&veaction=edit&section=3" title="Modifică secțiunea: Note" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice_cu_toate_elementele_1&action=edit&section=3" title="Edit section's source code: Note"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text">Tiberiu Vasile Trif, <a rel="nofollow" class="external text" href="http://math.ubbcluj.ro/~ttrif/Analiza2.pdf"><i>Analiză matematică</i></a>, Cluj-Napoca, Ed. Casa Cărții de Știință, 2017, <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/978-606-17-1102-4" title="Special:Referințe în cărți/978-606-17-1102-4">ISBN: 978-606-17-1102-4</a>, p. 18</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">Horn, Johnson, 2012, p. 8</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="Reference-Mathworld-Unit_Matrix"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Eric W. Weisstein</a>, <i><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/UnitMatrix.html">Unit Matrix</a></i> la <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite></span> </li> <li id="cite_note-S-4">^ <a href="#cite_ref-S_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-S_4-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="CITEREFStanley2013" class="citation">Stanley, Richard P. (<time datetime="2013">2013</time>), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_Tc_AAAAQBAJ&pg=PA4"><i>Algebraic Combinatorics: Walks, Trees, Tableaux, and More</i></a>, Springer, Lemma 1.4, p. 4, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/9781461469988" title="Special:Referințe în cărți/9781461469988">9781461469988</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Combinatorics%3A+Walks%2C+Trees%2C+Tableaux%2C+and+More&rft.pages=Lemma+1.4%2C+p.-4&rft.pub=Springer&rft.date=2013&rft.isbn=9781461469988&rft.aulast=Stanley&rft.aufirst=Richard+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_Tc_AAAAQBAJ%26pg%3DPA4&rfr_id=info%3Asid%2Fro.wikipedia.org%3AMatrice+cu+toate+elementele+1" class="Z3988"><span style="display:none;"> </span></span><style data-mw-deduplicate="TemplateStyles:r16236537">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"„""”""«""»"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}</style>.</span> </li> <li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text">Horn, Johnson, 2012, p. 65</span> </li> <li id="cite_note-timm-6">^ <a href="#cite_ref-timm_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-timm_6-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="CITEREFTimm2002" class="citation">Timm, Neil H. (<time datetime="2002">2002</time>), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vtiyg6fnnskC&pg=PA30"><i>Applied Multivariate Analysis</i></a>, Springer texts in statistics, Springer, p. 30, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/9780387227719" title="Special:Referințe în cărți/9780387227719">9780387227719</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Multivariate+Analysis&rft.series=Springer+texts+in+statistics&rft.pages=30&rft.pub=Springer&rft.date=2002&rft.isbn=9780387227719&rft.aulast=Timm&rft.aufirst=Neil+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dvtiyg6fnnskC%26pg%3DPA30&rfr_id=info%3Asid%2Fro.wikipedia.org%3AMatrice+cu+toate+elementele+1" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> <li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="CITEREFSmith2011" class="citation">Smith, Jonathan D. H. (<time datetime="2011">2011</time>), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PQUAQh04lrUC&pg=PA77"><i>Introduction to Abstract Algebra</i></a>, CRC Press, p. 77, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/9781420063721" title="Special:Referințe în cărți/9781420063721">9781420063721</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Abstract+Algebra&rft.pages=77&rft.pub=CRC+Press&rft.date=2011&rft.isbn=9781420063721&rft.aulast=Smith&rft.aufirst=Jonathan+D.+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPQUAQh04lrUC%26pg%3DPA77&rfr_id=info%3Asid%2Fro.wikipedia.org%3AMatrice+cu+toate+elementele+1" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537">.</span> </li> <li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="reference-text"><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="CITEREFGodsil1993" class="citation">Godsil, Chris (<time datetime="1993">1993</time>), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eADtlNCkkIMC&pg=PA25"><i>Algebraic Combinatorics</i></a>, CRC Press, Lemma 4.1, p. 25, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/9780412041310" title="Special:Referințe în cărți/9780412041310">9780412041310</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Combinatorics&rft.pages=Lemma+4.1%2C+p.-25&rft.pub=CRC+Press&rft.date=1993&rft.isbn=9780412041310&rft.aulast=Godsil&rft.aufirst=Chris&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeADtlNCkkIMC%26pg%3DPA25&rfr_id=info%3Asid%2Fro.wikipedia.org%3AMatrice+cu+toate+elementele+1" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografie">Bibliografie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Matrice_cu_toate_elementele_1&veaction=edit&section=4" title="Modifică secțiunea: Bibliografie" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Matrice_cu_toate_elementele_1&action=edit&section=4" title="Edit section's source code: Bibliografie"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span style="border:solid 1px #44A; background-color:#EEF; font-family:monospace; color:#008; font-size:0.9em; padding:0px 4px 2px 4px; position:relative; bottom:0.2em; cursor:help;" title="Limba engleză">en</span> <cite id="CITEREFHornJohnson2012" class="citation">Horn, Roger A.; Johnson, Charles R. (<time datetime="2012">2012</time>), „0.2.8 The all-ones matrix and vector”, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5I5AYeeh0JUC&pg=PA8"><i>Matrix Analysis</i></a>, Cambridge University Press, p. 8, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/9780521839402" title="Special:Referințe în cărți/9780521839402">9780521839402</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=0.2.8+The+all-ones+matrix+and+vector&rft.btitle=Matrix+Analysis&rft.pages=8&rft.pub=Cambridge+University+Press&rft.date=2012&rft.isbn=9780521839402&rft.aulast=Horn&rft.aufirst=Roger+A.&rft.au=Johnson%2C+Charles+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5I5AYeeh0JUC%26pg%3DPA8&rfr_id=info%3Asid%2Fro.wikipedia.org%3AMatrice+cu+toate+elementele+1" class="Z3988"><span style="display:none;"> </span></span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16236537"></li></ul> <div class="noprint tright portal" style="border:solid #aaa 1px; margin:0.5em 0 0.5em 0.5em;"> <table style="background:var(--background-color-interactive-subtle, #f9f9f9); color:inherit; font-size:85%; line-height:110%; max-width:175px;"> <tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Nuvola_apps_edu_mathematics-p-blue.svg" class="mw-file-description"><img alt="Portal icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_mathematics-p-blue.svg/28px-Nuvola_apps_edu_mathematics-p-blue.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_mathematics-p-blue.svg/42px-Nuvola_apps_edu_mathematics-p-blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Nuvola_apps_edu_mathematics-p-blue.svg/56px-Nuvola_apps_edu_mathematics-p-blue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </td> <td style="padding: 0 0.2em; vertical-align: middle; font-style: italic; font-weight: bold"><b><a href="/wiki/Portal:Matematic%C4%83" title="Portal:Matematică">Portal Matematică </a></b> </td></tr> </tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Adus de la <a dir="ltr" href="https://ro.wikipedia.org/w/index.php?title=Matrice_cu_toate_elementele_1&oldid=16054560">https://ro.wikipedia.org/w/index.php?title=Matrice_cu_toate_elementele_1&oldid=16054560</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categorii" title="Special:Categorii">Categorii</a>: <ul><li><a href="/wiki/Categorie:Matrice" title="Categorie:Matrice">Matrice</a></li><li><a href="/wiki/Categorie:1_(num%C4%83r)" title="Categorie:1 (număr)">1 (număr)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Ultima editare a paginii a fost efectuată la 16 ianuarie 2024, ora 20:55.</li> <li id="footer-info-copyright">Acest text este disponibil sub licența <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.ro">Creative Commons cu atribuire și distribuire în condiții identice</a>; pot exista și clauze suplimentare. 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