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mw-portlet-user-menu-anon-editor" lang="zh-Hans" dir="ltr"> <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="有关本IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-使用范例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#使用范例"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>使用范例</span> </div> </a> <ul id="toc-使用范例-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考资料" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考资料"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>参考资料</span> </div> </a> <ul id="toc-参考资料-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" lang="zh-Hans" dir="ltr"> <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading" lang="zh-Hans" dir="ltr"><span class="mw-page-title-main">增广矩阵</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" lang="zh-Hans" dir="ltr"> <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。20种语言可用" > <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-20" 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">20种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_%D9%85%D9%85%D8%AA%D8%AF%D8%A9" title="مصفوفة ممتدة – 阿拉伯语" lang="ar" hreflang="ar" data-title="مصفوفة ممتدة" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Pro%C5%A1irena_matrica" title="Proširena matrica – 波斯尼亚语" lang="bs" hreflang="bs" data-title="Proširena matrica" data-language-autonym="Bosanski" data-language-local-name="波斯尼亚语" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_ampliada" title="Matriu ampliada – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Matriu ampliada" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Roz%C5%A1%C3%AD%C5%99en%C3%A1_matice" title="Rozšířená matice – 捷克语" lang="cs" hreflang="cs" data-title="Rozšířená matice" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Totalmatrix" title="Totalmatrix – 丹麦语" lang="da" hreflang="da" data-title="Totalmatrix" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Erweiterte_Matrix" title="Erweiterte Matrix – 德语" lang="de" hreflang="de" data-title="Erweiterte Matrix" data-language-autonym="Deutsch" data-language-local-name="德语" 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%95%CF%80%CE%B1%CF%85%CE%BE%CE%B7%CE%BC%CE%AD%CE%BD%CE%BF%CF%82_%CF%80%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82" title="Επαυξημένος πίνακας – 希腊语" lang="el" hreflang="el" data-title="Επαυξημένος πίνακας" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" 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/Augmented_matrix" title="Augmented matrix – 英语" lang="en" hreflang="en" data-title="Augmented matrix" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matriz_aumentada" title="Matriz aumentada – 西班牙语" lang="es" hreflang="es" data-title="Matriz aumentada" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Matrize_zabaldu" title="Matrize zabaldu – 巴斯克语" lang="eu" hreflang="eu" data-title="Matrize zabaldu" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/T%C3%A4ydennetty_matriisi" title="Täydennetty matriisi – 芬兰语" lang="fi" hreflang="fi" data-title="Täydennetty matriisi" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8B%A1%E5%A4%A7%E8%A1%8C%E5%88%97" title="拡大行列 – 日语" lang="ja" hreflang="ja" data-title="拡大行列" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%A8%EA%B0%80_%ED%96%89%EB%A0%AC" title="첨가 행렬 – 韩语" lang="ko" hreflang="ko" data-title="첨가 행렬" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matriz_aumentada" title="Matriz aumentada – 葡萄牙语" lang="pt" hreflang="pt" data-title="Matriz aumentada" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Matrice_extins%C4%83" title="Matrice extinsă – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Matrice extinsă" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Raz%C5%A1irjena_matrika" title="Razširjena matrika – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Razširjena matrika" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%BF%E0%AE%B0%E0%AE%BF%E0%AE%B5%E0%AF%81%E0%AE%AA%E0%AE%9F%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9F%E0%AF%8D%E0%AE%9F_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="விரிவுபடுத்தப்பட்ட அணி – 泰米尔语" lang="ta" hreflang="ta" data-title="விரிவுபடுத்தப்பட்ட அணி" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" 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%B9%81%E0%B8%95%E0%B9%88%E0%B8%87%E0%B9%80%E0%B8%95%E0%B8%B4%E0%B8%A1" title="เมทริกซ์แต่งเติม – 泰语" lang="th" hreflang="th" data-title="เมทริกซ์แต่งเติม" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D1%88%D0%B8%D1%80%D0%B5%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F" title="Розширена матриця – 乌克兰语" lang="uk" hreflang="uk" data-title="Розширена матриця" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ma_tr%E1%BA%ADn_b%E1%BB%95_sung" title="Ma trận bổ sung – 越南语" lang="vi" hreflang="vi" data-title="Ma trận bổ sung" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2750301#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" lang="zh-Hans" dir="ltr"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5" rel="discussion" title="有关内容页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " lang="zh-Hans" dir="ltr"> <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="更改语言变体" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">简体</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" lang="zh-Hans" dir="ltr"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="ca-variants-zh mw-list-item"><a href="/zh/%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="selected ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5" lang="zh-Hant" 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</div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub" lang="zh-Hans" dir="ltr"><div id="mw-content-subtitle" lang="zh-Hans" dir="ltr"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh-Hans" dir="ltr"><div id="noteTA-307af9e9" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div></div> <table class="infobox noprint" style="width:210px; float: right; clear: right; text-align:center; margin-top:1em;"> <tbody><tr> <th style="font-size:90%; background:#DCF0FF"><a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a> </th></tr> <tr> <td><div style="padding-top: 7px; padding-bottom: 4px;"><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 \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31efc33ac33577d719a3ccd162a9bf21e4847ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.972ex; height:6.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}"></span></div> </td></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (线性代数)">基底</a> <span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a> <span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a></span> </td></tr> <tr> <td> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">向量 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E6%A0%87%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="标量 (数学)">标量</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">向量投影</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%A4%96%E7%A7%AF" class="mw-disambig" title="外积">外积</a>（<a href="/wiki/%E5%8F%89%E7%A7%AF" title="叉积">向量积</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%B8%83%E7%BB%B4%E5%8F%89%E7%A7%AF" title="七维叉积">七维向量积</a>） ·</span> <span class="nowrap"><a href="/wiki/%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4" title="内积空间">内积</a>（<a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">数量积</a>） ·</span> <span class="nowrap"><a href="/wiki/%E4%BA%8C%E9%87%8D%E5%90%91%E9%87%8F" title="二重向量">二重向量</a></span> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">矩阵与行列式 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="线性方程组">线性方程组</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%A7%A9_(%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0)" title="秩 (线性代数)">秩</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%9B%B6%E7%A9%BA%E9%97%B4" title="零空间">核</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%B7%A1" title="迹">迹</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%96%AE%E4%BD%8D%E7%9F%A9%E9%99%A3" title="单位矩阵">单位矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%88%9D%E7%AD%89%E7%9F%A9%E9%98%B5" title="初等矩阵">初等矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%96%B9%E5%9D%97%E7%9F%A9%E9%98%B5" title="方块矩阵">方块矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%88%86%E5%A1%8A%E7%9F%A9%E9%99%A3" title="分块矩阵">分块矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%B8%89%E8%A7%92%E7%9F%A9%E9%98%B5" title="三角矩阵">三角矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%9D%9E%E5%A5%87%E5%BC%82%E6%96%B9%E9%98%B5" title="非奇异方阵">非奇异方阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%BD%AC%E7%BD%AE%E7%9F%A9%E9%98%B5" title="转置矩阵">转置矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%80%86%E7%9F%A9%E9%98%B5" title="逆矩阵">逆矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%B0%8D%E8%A7%92%E7%9F%A9%E9%99%A3" title="对角矩阵">对角矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%8F%AF%E5%AF%B9%E8%A7%92%E5%8C%96%E7%9F%A9%E9%98%B5" title="可对角化矩阵">可对角化矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" title="对称矩阵">对称矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%8F%8D%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" title="反对称矩阵">反对称矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5" title="正交矩阵">正交矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%85%89%E7%9F%A9%E9%98%B5" title="酉矩阵">幺正矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="埃尔米特矩阵">埃尔米特矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%96%9C%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="斜埃尔米特矩阵">反埃尔米特矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E8%A7%84%E7%9F%A9%E9%98%B5" title="正规矩阵">正规矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%BC%B4%E9%9A%8F%E7%9F%A9%E9%98%B5" title="伴随矩阵">伴随矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%A4%98%E5%9B%A0%E5%AD%90%E7%9F%A9%E9%99%A3" title="馀因子矩阵">余因子矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%85%B1%E8%BD%AD%E8%BD%AC%E7%BD%AE" title="共轭转置">共轭转置</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5" title="正定矩阵">正定矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%B9%82%E9%9B%B6%E7%9F%A9%E9%98%B5" title="幂零矩阵">幂零矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3" title="矩阵分解">矩阵分解</a> （<a href="/wiki/LU%E5%88%86%E8%A7%A3" title="LU分解">LU分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3" title="奇异值分解">奇异值分解</a> ·</span> <span class="nowrap"><a href="/wiki/QR%E5%88%86%E8%A7%A3" title="QR分解">QR分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%9E%81%E5%88%86%E8%A7%A3" title="极分解">极分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%89%B9%E5%BE%81%E5%88%86%E8%A7%A3" title="特征分解">特征分解</a>） ·</span> <span class="nowrap"><a href="/wiki/%E5%AD%90%E5%BC%8F%E5%92%8C%E4%BD%99%E5%AD%90%E5%BC%8F" title="子式和余子式">子式和余子式</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%B1%95%E5%BC%80" title="拉普拉斯展开">拉普拉斯展开</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B%E7%A7%AF" title="克罗内克积">克罗内克积</a></span> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">线性空间与线性变换 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">线性空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性变换</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E5%AD%90%E7%A9%BA%E9%97%B4" title="线性子空间">线性子空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E7%94%9F%E6%88%90%E7%A9%BA%E9%97%B4" title="线性生成空间">线性生成空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (线性代数)">基</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性映射</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%8A%95%E5%BD%B1" class="mw-disambig" title="投影">线性投影</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%B7%9A%E6%80%A7%E7%84%A1%E9%97%9C" title="线性无关">线性无关</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E7%BB%84%E5%90%88" title="线性组合">线性组合</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%B3%9B%E5%87%BD" class="mw-redirect" title="线性泛函">线性泛函</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%A1%8C%E7%A9%BA%E9%97%B4%E4%B8%8E%E5%88%97%E7%A9%BA%E9%97%B4" title="行空间与列空间">行空间与列空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%AF%B9%E5%81%B6%E7%A9%BA%E9%97%B4" title="对偶空间">对偶空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E4%BA%A4" title="正交">正交</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%89%B9%E5%BE%81%E5%80%BC%E5%92%8C%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F" title="特征值和特征向量">特征向量</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95" title="最小二乘法">最小二乘法</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%A0%BC%E6%8B%89%E5%A7%86-%E6%96%BD%E5%AF%86%E7%89%B9%E6%AD%A3%E4%BA%A4%E5%8C%96" title="格拉姆-施密特正交化">格拉姆-施密特正交化</a></span> </td></tr></tbody></table> </td></tr> <tr style="text-align: center; font-size: 90%;"> <td><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:" :"}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist-pipe dd::after,.mw-parser-output .hlist-pipe li::after{content:" | ";font-weight:normal}.mw-parser-output .hlist-hyphen dd::after,.mw-parser-output .hlist-hyphen li::after{content:" - ";font-weight:normal}.mw-parser-output .hlist-comma dd::after,.mw-parser-output .hlist-comma li::after{content:"、";font-weight:normal}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:"（"counter(listitem)"\a0 "}.mw-parser-output ul.cslist,.mw-parser-output ul.sslist{margin:0;padding:0;display:inline-block;list-style:none}.mw-parser-output .cslist li,.mw-parser-output .sslist li{margin:0;display:inline-block}.mw-parser-output .cslist li::after{content:"，"}.mw-parser-output .sslist li::after{content:"；"}.mw-parser-output .cslist li:last-child::after,.mw-parser-output .sslist li:last-child::after{content:none}</style><style data-mw-deduplicate="TemplateStyles:r84244141">.mw-parser-output .navbar{display:inline;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="Template:线性代数"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0&action=edit&redlink=1" class="new" title="Template talk:线性代数（页面不存在）"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="Special:编辑页面/Template:线性代数"><abbr title="编辑该模板">编</abbr></a></li></ul></div> </td></tr></tbody></table> <p><b>增广矩阵</b>，又称<b>广置矩阵</b>，是在<a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a>中系数<a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>的右边添上<a href="/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="线性方程组">线性方程组</a>等号右边的<a href="/wiki/%E5%B8%B8%E6%95%B0" title="常数">常数</a>列得到的矩阵，如：<a href="/wiki/%E6%96%B9%E7%A8%8B" title="方程">方程</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 AX=B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>X</mi> <mo>=</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AX=B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e51b626e9793be3e577dc5157905281ec0af0d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.586ex; height:2.176ex;" alt="{\displaystyle AX=B}"></span>系数矩阵为<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><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}"></span>，它的增广矩阵为<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|B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A|B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db461fcd5c69180cd914dd6ce977f95cdffb9fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.963ex; height:2.843ex;" alt="{\displaystyle (A|B)}"></span>。<a href="/wiki/%E6%96%B9%E7%A8%8B%E7%BB%84" title="方程组">方程组</a>唯一确定增广矩阵，通过增广矩阵的<a href="/wiki/%E5%88%9D%E7%AD%89%E8%A1%8C%E8%AE%8A%E6%8F%9B" class="mw-redirect" title="初等行变换">初等行变换</a>可用于判断对应线性方程组是否有<a href="/wiki/%E6%96%B9%E7%A8%8B%E6%B1%82%E8%A7%A3" title="方程求解">解</a>，以及化简求原方程组的解。 </p> <div class="mw-heading mw-heading2"><h2 id="使用范例"><span id=".E4.BD.BF.E7.94.A8.E8.8C.83.E4.BE.8B"></span>使用范例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5&action=edit&section=1" title="编辑章节：使用范例"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>增广矩阵通常用于判断矩阵的有解的情况，下列<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><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}"></span>为线性方程组的系数矩阵，<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|B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A|B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db461fcd5c69180cd914dd6ce977f95cdffb9fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.963ex; height:2.843ex;" alt="{\displaystyle (A|B)}"></span>为增广矩阵： </p> <ul><li>若<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 \operatorname {rank} (A)<\operatorname {rank} (A|B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rank} (A)<\operatorname {rank} (A|B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e3c79a8e9ea013acd5ed7f9f036404d8239c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.802ex; height:2.843ex;" alt="{\displaystyle \operatorname {rank} (A)<\operatorname {rank} (A|B)}"></span>，方程组无解。</li> <li>若<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 \operatorname {rank} (A)=\operatorname {rank} (A|B)=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rank} (A)=\operatorname {rank} (A|B)=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6771b6f71a9dea7683715659c00226730f38f948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.295ex; height:2.843ex;" alt="{\displaystyle \operatorname {rank} (A)=\operatorname {rank} (A|B)=n}"></span>，方程组有唯一解。</li> <li>若<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 \operatorname {rank} (A)=\operatorname {rank} (A|B)<n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rank} (A)=\operatorname {rank} (A|B)<n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/616636d18720f492d3f8a5bc253ada5f7fd465dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.295ex; height:2.843ex;" alt="{\displaystyle \operatorname {rank} (A)=\operatorname {rank} (A|B)<n}"></span>，方程组无穷解。</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 \operatorname {rank} (A)>\operatorname {rank} (A|B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {rank} (A)>\operatorname {rank} (A|B)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0899b22d94cc7f25e6b9e7be9d9fabfc06fb1e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.802ex; height:2.843ex;" alt="{\displaystyle \operatorname {rank} (A)>\operatorname {rank} (A|B)}"></span>不会发生，因为增广矩阵的<a href="/wiki/%E7%A7%A9_(%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0)" title="秩 (线性代数)">秩</a>大于等于系数矩阵的秩。</li></ul> <p>对于如下方程组： </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{cases}x+2y+3z=0\\3x+4y+7z=2\\6x+5y+9z=11\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mi>z</mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mi>y</mi> <mo>+</mo> <mn>7</mn> <mi>z</mi> <mo>=</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mi>y</mi> <mo>+</mo> <mn>9</mn> <mi>z</mi> <mo>=</mo> <mn>11</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}x+2y+3z=0\\3x+4y+7z=2\\6x+5y+9z=11\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11611db6f76180b879f3006bd84afa391c8b40d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:20.982ex; height:8.509ex;" alt="{\displaystyle {\begin{cases}x+2y+3z=0\\3x+4y+7z=2\\6x+5y+9z=11\end{cases}}}"></span></dd></dl> <p>系数矩阵为： </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 A={\begin{bmatrix}1&2&3\\3&4&7\\6&5&9\end{bmatrix}},B={\begin{bmatrix}0\\2\\11\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>9</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>11</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}1&2&3\\3&4&7\\6&5&9\end{bmatrix}},B={\begin{bmatrix}0\\2\\11\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79833d70b197c7f1f875a411c137e8e7d154f319" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:28.9ex; height:9.176ex;" alt="{\displaystyle A={\begin{bmatrix}1&2&3\\3&4&7\\6&5&9\end{bmatrix}},B={\begin{bmatrix}0\\2\\11\end{bmatrix}}}"></span></dd></dl> <p>增广矩阵为： </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 (A|B)=\left[{\begin{array}{ccc|c}1&2&3&0\\3&4&7&2\\6&5&9&11\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>7</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>11</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A|B)=\left[{\begin{array}{ccc|c}1&2&3&0\\3&4&7&2\\6&5&9&11\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fe6dc02ad503bbb9878db909bfd1ed3429fcb78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:27.552ex; height:10.176ex;" alt="{\displaystyle (A|B)=\left[{\begin{array}{ccc|c}1&2&3&0\\3&4&7&2\\6&5&9&11\end{array}}\right]}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="参考资料"><span id=".E5.8F.82.E8.80.83.E8.B5.84.E6.96.99"></span>参考资料</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5&action=edit&section=2" title="编辑章节：参考资料"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>同济大学数学系．工程数学线性代数（第五版）．北京市西城区德外大街4号：高等教育出版社，2011-11：P64．</li></ul> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5&action=edit&section=3" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E9%AB%98%E6%96%AF%E6%B6%88%E5%85%83%E6%B3%95" class="mw-redirect" title="高斯消元法">高斯消元法</a></li></ul>    </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="" lang="zh-Hans" 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