{"id":294,"date":"2023-07-06T17:52:46","date_gmt":"2023-07-06T17:52:46","guid":{"rendered":"https:\/\/mathority.org\/de\/beispiele-zum-losen-von-matrixgleichungen-und-geloste-ubungen-zu-2x2-und-3x3-matrizen\/"},"modified":"2023-07-06T17:52:46","modified_gmt":"2023-07-06T17:52:46","slug":"beispiele-zum-losen-von-matrixgleichungen-und-geloste-ubungen-zu-2x2-und-3x3-matrizen","status":"publish","type":"post","link":"https:\/\/mathority.org\/de\/beispiele-zum-losen-von-matrixgleichungen-und-geloste-ubungen-zu-2x2-und-3x3-matrizen\/","title":{"rendered":"Matrixgleichungen"},"content":{"rendered":"<p>Auf dieser Seite erfahren Sie, was <strong>Matrixgleichungen<\/strong> sind und wie Sie sie l\u00f6sen. Dar\u00fcber hinaus finden Sie Beispiele und gel\u00f6ste Aufgaben zu Gleichungen mit Matrizen.<\/p>\n<h2 class=\"wp-block-heading\"> Was sind Matrixgleichungen?<\/h2>\n<p> <strong>Matrixgleichungen<\/strong> sind wie normale Gleichungen, bestehen jedoch nicht aus Zahlen, sondern aus Matrizen. Zum Beispiel:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2bc59624603d48ea9b4df50b4c052437_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  AX=B\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Daher wird L\u00f6sung X auch eine Matrix sein.<\/p>\n<p> Wie Sie bereits wissen, k\u00f6nnen Matrizen nicht geteilt werden. Daher kann die Matrix X nicht durch Division der Matrix, die sie multipliziert hat, auf der anderen Seite der Gleichung gel\u00f6scht werden:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-13f935eb2129110be40aa176554bb557_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\renewcommand{\\CancelColor}{\\color{red}}  \\xcancel{X =\\cfrac{B}{A}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"57\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Im Gegenteil, um die X-Matrix zu l\u00f6schen, muss ein ganzes Verfahren befolgt werden. Sehen wir uns also an, wie man Matrixgleichungen mit einer gel\u00f6sten \u00dcbung l\u00f6st:<\/p>\n<h2 class=\"estil_titol_H2 wp-block-heading\"> So l\u00f6sen Sie Matrixgleichungen. Beispiel:<\/h2>\n<ul>\n<li> L\u00f6sen Sie die folgende Matrixgleichung:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9274aedf7d1f424b7e21547f7968321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  AX+B = C\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9727c78818a9661573310f22ec2fb3cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A =\\begin{pmatrix}2 &amp; 1 \\\\[1.1ex] 4 &amp; 3 \\end{pmatrix} \\qquad B = \\begin{pmatrix} 3 &amp; -1 \\\\[1.1ex] 0 &amp; 5 \\end{pmatrix} \\qquad C =\\begin{pmatrix} 2 &amp; 1 \\\\[1.1ex] 6 &amp; -3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"399\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Als Erstes m\u00fcssen wir nach Matrix X aufl\u00f6sen <strong>. Also subtrahieren wir Matrix B<\/strong> <strong>von der anderen Seite der Gleichung:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9274aedf7d1f424b7e21547f7968321_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  AX+B = C\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e74a9aaf9e3c11fb261374224402346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  AX = C-B\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"103\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Zum Abschluss kann die Clearing-Matrix nicht geteilt werden. Aber wir m\u00fcssen Folgendes tun:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Wir m\u00fcssen beide Seiten der Gleichung mit der <strong>Umkehrung der Matrix multiplizieren, die die Matrix X multipliziert,<\/strong> und au\u00dferdem beide Seiten <strong>mit der Seite multiplizieren, auf der sich diese Matrix befindet.<\/strong><\/p>\n<p> In diesem Fall ist die Matrix, die X multipliziert, A und befindet sich links davon. <strong>Wir multiplizieren daher links beide Seiten der Gleichung mit der Umkehrung von A<\/strong> (A <sup>-1<\/sup> ):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e74a9aaf9e3c11fb261374224402346_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  AX = C-B\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"103\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d233c41796c59c73995600f80e74f323_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\definecolor{vermell}{HTML}{F44336} \\color{vermell}\\bm{A^{-1}} \\color{black} \\cdot AX =  \\color{vermell}\\bm{A^{-1}} \\color{black}  \\cdot (C-B)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"587\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Eine mit ihrer Umkehrung multiplizierte Matrix ist gleich der Identit\u00e4tsmatrix. Noch<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7ecd5173741978b59218941381221723_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{A^{-1} \\cdot A = I }:\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"100\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-58279bb023cd9b14c2019eccfc240afa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  IX = A^{-1} \\cdot (C-B)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Jede mit der Identit\u00e4tsmatrix multiplizierte Matrix ergibt dieselbe Matrix. Noch:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de90329d45b7fa427640506649c111e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  X = A^{-1} \\cdot (C-B)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Und auf diese Weise <strong>haben wir X bereits gel\u00f6scht.<\/strong> F\u00fchren Sie nun nur noch die Matrixoperationen durch. Also berechnen wir zun\u00e4chst die <a href=\"https:\/\/mathority.org\/de\/inverse-matrix\/\">2 \u00d7 2-Umkehrmatrix<\/a> von A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b79c0ae6349ac5ac0267e179e641b66e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A =\\begin{pmatrix}2 &amp; 1 \\\\[1.1ex] 4 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1fe85ec6c4385daba7d2488b0d60ee2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1} = \\cfrac{1}{\\vert A \\vert } \\cdot \\Bigl( \\text{Adj}(A)\\Bigr)^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"175\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Wir berechnen den Adjungierten der Matrix A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1eb7c7a828453c5310d59386f0303b83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\cfrac{1}{2} \\cdot \\begin{pmatrix}3 &amp; -4 \\\\[1.1ex] -1 &amp; 2 \\end{pmatrix}^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"173\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Und sobald die adjungierte Matrix gefunden ist, fahren wir mit der Berechnung der <a href=\"https:\/\/mathority.org\/de\/beispiele-fur-transponierte-oder-transponierte-matrix-und-geloste-ubungen\/\">transponierten Matrix<\/a> fort, um die inverse Matrix zu bestimmen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa12c355319a6894e76343c9cb9185d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\cfrac{1}{2} \\cdot \\begin{pmatrix}3 &amp; -1 \\\\[1.1ex] -4 &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"164\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2fd06e0ad4a2a18560f644b718dadf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\begin{pmatrix} \\frac{3}{2} &amp; -\\frac{1}{2} \\\\[1.3ex] -2 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"143\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Jetzt setzen wir alle Matrizen in den Ausdruck ein, um X zu berechnen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de90329d45b7fa427640506649c111e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  X = A^{-1} \\cdot (C-B)\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-99716e9accb7ee578fb1119d4e800e4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  X = \\begin{pmatrix} \\frac{3}{2} &amp; -\\frac{1}{2} \\\\[1.3ex] -2 &amp; 1\\end{pmatrix} \\cdot \\left(\\begin{pmatrix} \\vphantom{\\frac{3}{2}} 2 &amp; 1 \\\\[1.3ex] 6 &amp; -3\\end{pmatrix}-\\begin{pmatrix} \\vphantom{\\frac{3}{2}}3 &amp; -1 \\\\[1.3ex] 0 &amp; 5 \\end{pmatrix}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"341\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Und wir fahren damit fort, die Operationen mit Matrizen zu l\u00f6sen. Wir berechnen zun\u00e4chst die Klammern, indem wir die Matrizen subtrahieren:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d07c28ad6104e391605836ecdd297251_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  X = \\begin{pmatrix} \\frac{3}{2} &amp; -\\frac{1}{2} \\\\[1.3ex] -2 &amp; 1\\end{pmatrix}\\begin{pmatrix} -1 &amp; 2 \\\\[1.1ex] 6 &amp; -8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"220\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Und schlie\u00dflich multiplizieren wir die Matrizen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b28076f6ab18dc77a0083388046c5cd1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  X = \\begin{pmatrix} \\frac{3}{2}\\cdot (-1) + \\left(-\\frac{1}{2} \\right) \\cdot 6 &amp; \\frac{3}{2}\\cdot 2 + \\left(-\\frac{1}{2} \\right)\\cdot (-8) \\\\[1.3ex] -2\\cdot (-1)+1\\cdot 6 &amp; -2\\cdot 2 +1\\cdot (-8) \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"368\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7d20e85150a382ba9f11bf328b866834_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  X = \\begin{pmatrix} -\\frac{3}{2} -\\frac{6}{2} &amp; 3 + 4 \\\\[1.3ex] 2+6 &amp; -4-8 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"190\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5d3e7ebae094a92690d97b614b0487a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{X =} \\begin{pmatrix} \\bm{-} \\frac{\\bm{9}}{\\bm{2}} &amp; \\bm{7} \\\\[1.3ex] \\bm{8} &amp; \\bm{-12} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"134\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Probleme mit Matrixgleichungen gel\u00f6st<\/h2>\n<p> Damit Sie das Konzept \u00fcben und somit gut verstehen k\u00f6nnen, hinterlassen wir Ihnen im Folgenden einige gel\u00f6ste Matrixgleichungen. Sie k\u00f6nnen die \u00dcbungen ausprobieren und sehen, ob Ihnen die L\u00f6sungen gelungen sind. Vergessen Sie nicht, dass Sie uns auch alle Fragen stellen k\u00f6nnen, die in den Kommentaren auftauchen.<\/p>\n<h3 class=\"wp-block-heading\"> \u00dcbung 1<\/h3>\n<p> Sei<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36386dbc4f20fb573357a406ce713887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a08b2dd56803fba7d8e5a0dcb0430601_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> die folgenden quadratischen Matrizen der Dimension 2\u00d72:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0f40b96fc0f1047fb0c39a7d41be04ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A =\\begin{pmatrix} 3 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} \\qquad B = \\begin{pmatrix} 4 &amp; 2 \\\\[1.1ex] -1 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"261\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Berechnen Sie die Matrix<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> welches die folgende Matrixgleichung erf\u00fcllt: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ec1e9c04147230526534e694fb54f316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle AX=B\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Sie m\u00fcssen zuerst die Matrix leeren<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> der Matrixgleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ec1e9c04147230526534e694fb54f316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle AX=B\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e79aa1830295bd486a911b5f5c279c9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1} \\cdot AX=A^{-1} \\cdot B\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"156\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d76831ec8e157e150f59ce0900114b77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle IX=A^{-1} \\cdot B\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"107\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc068e1794d487229ee0be3976454154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=A^{-1} \\cdot B\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sobald wir die Matrix haben<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> Klar, operiere einfach mit den Matrizen. Wir berechnen daher zun\u00e4chst die inverse Matrix von A: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fb5c4785b78010fcac56e1189338b99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A =\\begin{pmatrix} 3 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1fe85ec6c4385daba7d2488b0d60ee2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1} = \\cfrac{1}{\\vert A \\vert } \\cdot \\Bigl( \\text{Adj}(A)\\Bigr)^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"175\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7c4d4a6bfca6d2eedde52937c8ee0917_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\cfrac{1}{1} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 3 \\end{pmatrix}^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"160\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-695a05e4176ced4a4beaec27ce201b4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\cfrac{1}{1} \\cdot \\begin{pmatrix}0 &amp; 1 \\\\[1.1ex] -1 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"151\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a12ae8d0ae9ce16f04540ecd1a0ac907_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\begin{pmatrix} 0 &amp; 1 \\\\[1.1ex] -1 &amp; 3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"127\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jetzt ersetzen wir alle Matrizen in der Gleichung, um die Matrix zu berechnen <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f31b6ad36b8ba2d917f13bb377de636f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc068e1794d487229ee0be3976454154_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=A^{-1} \\cdot B\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-92d5f580fddfc830181cde2e67013987_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X= \\begin{pmatrix} 0 &amp; 1 \\\\[1.1ex] -1 &amp; 3\\end{pmatrix}\\cdot \\begin{pmatrix} 4 &amp; 2 \\\\[1.1ex] -1 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"200\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und schlie\u00dflich f\u00fchren wir die Multiplikation der Matrizen durch: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-787643b41cb362e276b8f80c9211fb52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{X=} \\begin{pmatrix}\\bm{ -1} &amp; \\bm{3} \\\\[1.1ex] \\bm{-7} &amp; \\bm{7}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 2<\/h3>\n<p> Sei<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36386dbc4f20fb573357a406ce713887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a08b2dd56803fba7d8e5a0dcb0430601_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cbff3f75ba97791e8db3213060854130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle C\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> die folgende Reihenfolge 2 Matrizen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f4f1f244d15039c64282a9fe347cee4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A =\\begin{pmatrix} 3 &amp; 6 \\\\[1.1ex] 2 &amp; -1 \\end{pmatrix} \\qquad B = \\begin{pmatrix} -2 &amp; 1 \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}\\qquad C = \\begin{pmatrix} 6 &amp; 4 \\\\[1.1ex] 3 &amp; -2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"426\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Berechnen Sie die Matrix<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> welches die folgende Matrixgleichung erf\u00fcllt: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2779ee661e4a42242acbed40277bf774_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A+ XB=C\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Als erstes m\u00fcssen wir die Matrix leeren.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> der Matrixgleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2779ee661e4a42242acbed40277bf774_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A+ XB=C\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0d8aea6239fb382563c5f5135145a77b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  XB=C-A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"103\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f7f75794139db4c21b8c91bb459a7a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle XB \\cdot B^{-1}=\\left(C-A\\right)\\cdot B^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"206\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8766c612738778657de57a198fb0cd29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle XI=\\left(C-A\\right)\\cdot B^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7bd40901bc80289bb49d0fd47f6236c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X = \\left(C-A\\right)\\cdot B^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sobald wir die Matrix isoliert haben<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> , ist es notwendig, mit Matrizen zu arbeiten. Wir berechnen daher zun\u00e4chst die inverse Matrix von B: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33c4a446ecdc391935728843e6a34964_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  B =\\begin{pmatrix} -2 &amp; 1 \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"123\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da88dade6c0344edc4f87207bc9b915c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B^{-1} = \\cfrac{1}{\\vert B \\vert } \\cdot \\Bigl( \\text{Adj}(B)\\Bigr)^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"178\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4850852b0e29a3d530b32dc1cd635499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  B^{-1} = \\cfrac{1}{3} \\cdot \\begin{pmatrix} -3 &amp; -3 \\\\[1.1ex] -1 &amp; -2 \\end{pmatrix}^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"174\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8817da5e89bc39e89bd17390cfd61c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  B^{-1} = \\cfrac{1}{3} \\cdot \\begin{pmatrix} -3 &amp; -1 \\\\[1.1ex] -3 &amp; -2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"165\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a5fc342354f6410cb87fa6b0ddf833a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  B^{-1} = \\begin{pmatrix} -1 &amp; -\\frac{1}{3} \\\\[1.3ex] -1 &amp; -\\frac{2}{3} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"144\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jetzt ersetzen wir alle Matrizen in der Gleichung, um die Matrix zu berechnen <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f31b6ad36b8ba2d917f13bb377de636f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2bda3fe0c2275283c3ce9dcd7cdfce4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\left(C-A\\right)\\cdot B^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"147\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-79abf2abf8a29e6357f65a1b62c9a80f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  X=\\left(\\begin{pmatrix} 6 &amp; 4 \\\\[1.3ex] 3 &amp; -2 \\end{pmatrix}-\\begin{pmatrix} 3 &amp; 6 \\\\[1.3ex] 2 &amp; -1 \\end{pmatrix}\\right)\\cdot \\begin{pmatrix} -1 &amp; -\\frac{1}{3} \\\\[1.3ex] -1 &amp; -\\frac{2}{3} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"341\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Wir l\u00f6sen die Klammern durch Subtrahieren der Matrizen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5141a4cb61be8db15676e185b10767f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} 3 &amp; -2 \\\\[1.3ex] 1 &amp; -1 \\end{pmatrix}\\cdot \\begin{pmatrix} -1 &amp; -\\frac{1}{3} \\\\[1.3ex] -1 &amp; -\\frac{2}{3} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und schlie\u00dflich multiplizieren wir die Matrizen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d5482b1eb8fd6af1d6c61547b05c0bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} -3+2 &amp; -1+\\frac{4}{3} \\\\[1.3ex] -1+1 &amp; -\\frac{1}{3}+\\frac{2}{3} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"190\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-779a021183e139f0e138fbc288d4adea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{X=} \\begin{pmatrix}\\bm{ -1} &amp; \\frac{\\bm{1}}{\\bm{3}} \\\\[1.3ex] \\bm{0} &amp; \\frac{\\bm{1}}{\\bm{3}} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"111\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> \u00dcbung 3<\/h3>\n<p> Sei<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36386dbc4f20fb573357a406ce713887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a08b2dd56803fba7d8e5a0dcb0430601_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cbff3f75ba97791e8db3213060854130_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle C\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> die folgenden Matrizen zweiter Ordnung:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6292882d305055e4e8fb287a4bc93b71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A =\\begin{pmatrix} -1 &amp; 1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} \\qquad B = \\begin{pmatrix} 4 &amp; -2 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix}\\qquad C = \\begin{pmatrix} 6 &amp; 4 \\\\[1.1ex] 22 &amp; 14 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"416\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Finden Sie die Matrix<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> welches die folgende Matrixgleichung erf\u00fcllt: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3241bccca1a61191660195f8076bb990_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle AXB=C\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"81\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Zuerst m\u00fcssen wir die Matrix l\u00f6schen<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> der Matrixgleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3241bccca1a61191660195f8076bb990_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle AXB=C\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"81\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1e90337c58e38fc6ec3c2b2c884d7fed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1}\\cdot AXB\\cdot B^{-1}=A^{-1}\\cdot C\\cdot B^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"260\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b4b610cde38418d268b1f5c5d01463d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displastyle IXI=A^{-1}\\cdot C\\cdot B^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"161\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e9076f1a9803ae41329636d95a8c8182_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displastyle X=A^{-1}\\cdot C\\cdot B^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"142\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sobald wir die Matrix geleert haben<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> , ist es notwendig, mit Matrizen zu arbeiten. Wir berechnen daher zun\u00e4chst die inverse Matrix von A: <\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b09ce42998b548267e70e47b135b6508_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A =\\begin{pmatrix} -1 &amp; 1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1fe85ec6c4385daba7d2488b0d60ee2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1} = \\cfrac{1}{\\vert A \\vert } \\cdot \\Bigl( \\text{Adj}(A)\\Bigr)^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"175\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a29b310de613bc1ec42a6e1452db147_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\cfrac{1}{-1} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] -1 &amp; -1 \\end{pmatrix}^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"187\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c0e0b895fed20ba908417f6ee3482ce0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\cfrac{1}{-1} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] -1 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"178\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c19685457cd40098cadf6eeff41405d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\begin{pmatrix} 0 &amp; 1 \\\\[1.1ex] 1 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"113\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und wir invertieren auch Matrix B: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3f5048394796b2378c8197c9c9c1cb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  B =\\begin{pmatrix} 4 &amp; -2 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"110\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da88dade6c0344edc4f87207bc9b915c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B^{-1} = \\cfrac{1}{\\vert B \\vert } \\cdot \\Bigl( \\text{Adj}(B)\\Bigr)^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"178\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-261eb432e305f5df596fc1dff9f183d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  B^{-1} = \\cfrac{1}{2} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 2 &amp; 4 \\end{pmatrix}^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"161\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96d40ae8aa7c350c8a63d57d06b6fa6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  B^{-1} = \\cfrac{1}{2} \\cdot \\begin{pmatrix} 0 &amp; 2 \\\\[1.1ex] -1 &amp; 4 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"152\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-80ee47f61b0671b42f9df06e7f384847_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  B^{-1} = \\begin{pmatrix} 0 &amp; 1 \\\\[1.3ex] -\\frac{1}{2} &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"130\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jetzt setzen wir alle Matrizen in den Ausdruck ein, um die Matrix zu berechnen <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f31b6ad36b8ba2d917f13bb377de636f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"25\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f765275df3d12633f97c500c3d7ca336_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=A^{-1}\\cdot C\\cdot B^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"142\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de94e47503b17f761f7fcb764f4def59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} 0 &amp; 1 \\\\[1.3ex] 1 &amp; 1 \\end{pmatrix}\\cdot\\begin{pmatrix} 6 &amp; 4 \\\\[1.3ex] 22 &amp; 14 \\end{pmatrix}\\cdot \\begin{pmatrix} 0 &amp; 1 \\\\[1.3ex] -\\frac{1}{2} &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"281\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Wir l\u00f6sen zun\u00e4chst die Multiplikation auf der linken Seite <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ca95f4870d5be13a3f7e241e5a40934b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} 0+22 &amp; 0+14 \\\\[1.3ex] 6+22 &amp; 4+14 \\end{pmatrix}\\cdot \\begin{pmatrix} 0 &amp; 1 \\\\[1.3ex] -\\frac{1}{2} &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"267\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3df7709b9d5c5f5194744d4c88d2cb66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} 22 &amp; 14 \\\\[1.3ex] 28 &amp; 18 \\end{pmatrix}\\cdot \\begin{pmatrix} 0 &amp; 1 \\\\[1.3ex] -\\frac{1}{2} &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"206\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und schlie\u00dflich f\u00fchren wir die verbleibende Multiplikation durch: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-62d83b02b8768a7e95ee71b7782d7759_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} 0-7 &amp; 22+28 \\\\[1.3ex] 0-9 &amp; 28+36 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"176\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b3a393915b3c49bdf9dd9ee6ada5020_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{X=} \\begin{pmatrix}\\bm{-7} &amp; \\bm{50} \\\\[1.3ex] \\bm{-9} &amp; \\bm{64} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"118\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">\u00dcbung 4<\/h3>\n<p> Sei<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36386dbc4f20fb573357a406ce713887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> Und<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a08b2dd56803fba7d8e5a0dcb0430601_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> die folgenden Matrizen der Dimension 3\u00d73:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da8b3d05ecc85eea72fd7d14c282f58c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A =\\begin{pmatrix}1 &amp; 0 &amp; 1\\\\[1.1ex] 0 &amp; -1 &amp; 0 \\\\[1.1ex] 1 &amp; 2 &amp; 2 \\end{pmatrix} \\qquad B = \\begin{pmatrix} 1 &amp; -1 &amp; 0 \\\\[1.1ex] 2 &amp; 3 &amp; -2 \\\\[1.1ex] -3 &amp; 1 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"344\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Berechnen Sie die Matrix<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> welches die folgende Matrixgleichung erf\u00fcllt: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-380b29380ba0a0dab3c183ea8b29e098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B^{t}- AX=B\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>siehe L\u00f6sung<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Zuerst l\u00f6schen wir die Matrix<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> der Matrixgleichung: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a038bf0cecb3080614f71975c72a41c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B^t- AX=B\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea7312ca34bee43be5f7727bdcf3ad3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B^t- B=AX\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"109\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9531df9bff4bb2e5a0015f0aa4c91d6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1}\\cdot \\left(B^t- B \\right)=A^{-1}\\cdot AX\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"214\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ea87eb7eca24cc40f458bb082b5bd0ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1}\\cdot \\left(B^t- B \\right)=IX\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"166\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-32d5781c515b740e3b7c20b62215d5bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1}\\cdot \\left(B^t- B \\right)=X\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"156\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6081c855462d0193b955600b1d5db48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=A^{-1}\\cdot \\left(B^t- B \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"154\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sobald wir die Matrix isoliert haben<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4ee28752517d6062a3ca0314890342d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"X\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"><\/p>\n<p> , ist es notwendig, mit Matrizen zu arbeiten. Wir berechnen daher zun\u00e4chst die inverse Matrix von A: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1a92fa898838b531bf1b51356dbbb2de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A =\\begin{pmatrix} 1 &amp; 0 &amp; 1\\\\[1.1ex] 0 &amp; -1 &amp; 0 \\\\[1.1ex] 1 &amp; 2 &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1fe85ec6c4385daba7d2488b0d60ee2d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1} = \\cfrac{1}{\\vert A \\vert } \\cdot \\Bigl( \\text{Adj}(A)\\Bigr)^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"175\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9cc1a5bb552d5eadacef8677265cba0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\cfrac{1}{-1} \\cdot \\begin{pmatrix} \\begin{vmatrix} -1 &amp; 0 \\\\ 2 &amp; 2 \\end{vmatrix} &amp; -\\begin{vmatrix} 0 &amp; 0 \\\\  1 &amp; 2 \\end{vmatrix} &amp; \\begin{vmatrix}  0 &amp; -1  \\\\ 1 &amp; 2 \\end{vmatrix}\\\\[4ex] -\\begin{vmatrix}  0 &amp; 1 \\\\ 2 &amp; 2 \\end{vmatrix} &amp; \\begin{vmatrix} 1  &amp; 1\\\\ 1 &amp; 2 \\end{vmatrix} &amp; -\\begin{vmatrix} 1 &amp; 0 \\\\ 1 &amp; 2  \\end{vmatrix} \\\\[4ex] \\begin{vmatrix} 0 &amp; 1\\\\  -1 &amp; 0 \\end{vmatrix} &amp; -\\begin{vmatrix} 1  &amp; 1\\\\ 0 &amp; 0  \\end{vmatrix} &amp; \\begin{vmatrix} 1 &amp; 0 \\\\ 0 &amp; -1 \\end{vmatrix} \\end{pmatrix}^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"175\" width=\"349\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e668ed3a6e233bed8245f99e80638633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\cfrac{1}{-1} \\cdot \\begin{pmatrix} -2 &amp; 0 &amp; 1 \\\\[1.1ex] 2 &amp; 1 &amp; -2 \\\\[1.1ex] 1  &amp; 0 &amp; -1 \\end{pmatrix}^{\\bm{t}}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"215\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eeb999734b9ba4b6e9a01e788bee6649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = -1 \\cdot \\begin{pmatrix} -2 &amp; 2 &amp; 1 \\\\[1.1ex] 0 &amp; 1 &amp; 0 \\\\[1.1ex] 1  &amp; -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"218\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f8f2379d6d616b29b78005aaafe39f29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A^{-1} = \\begin{pmatrix} 2 &amp; -2 &amp; -1 \\\\[1.1ex] 0 &amp; -1 &amp; 0 \\\\[1.1ex] -1  &amp; 2 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"182\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jetzt setzen wir alle Matrizen in den Ausdruck ein, um X zu berechnen: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6081c855462d0193b955600b1d5db48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=A^{-1}\\cdot \\left(B^t- B \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"154\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c91b944756316c7cde33eb90743d54d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} 2 &amp; -2 &amp; -1 \\\\[1.1ex] 0 &amp; -1 &amp; 0 \\\\[1.1ex] -1  &amp; 2 &amp; 1 \\end{pmatrix}\\cdot \\left(\\begin{pmatrix} 1 &amp; -1 &amp; 0 \\\\[1.1ex] 2 &amp; 3 &amp; -2 \\\\[1.1ex] -3 &amp; 1 &amp; -1 \\end{pmatrix}^t- \\begin{pmatrix} 1 &amp; -1 &amp; 0 \\\\[1.1ex] 2 &amp; 3 &amp; -2 \\\\[1.1ex] -3 &amp; 1 &amp; -1 \\end{pmatrix} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"501\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Wir transponieren Matrix B:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a81f0c3d7367d756d53221e9c56d1e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} 2 &amp; -2 &amp; -1 \\\\[1.1ex] 0 &amp; -1 &amp; 0 \\\\[1.1ex] -1  &amp; 2 &amp; 1 \\end{pmatrix}\\cdot \\left(\\begin{pmatrix} 1 &amp; 2 &amp; -3 \\\\[1.1ex] -1 &amp; 3 &amp; 1 \\\\[1.1ex] 0 &amp; -2 &amp; -1 \\end{pmatrix}- \\begin{pmatrix} 1 &amp; -1 &amp; 0 \\\\[1.1ex] 2 &amp; 3 &amp; -2 \\\\[1.1ex] -3 &amp; 1 &amp; -1 \\end{pmatrix} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"495\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Wir l\u00f6sen die Klammern durch Subtrahieren von Matrizen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5f822e84288230368a5c0918c79398bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle X=\\begin{pmatrix} 2 &amp; -2 &amp; -1 \\\\[1.1ex] 0 &amp; -1 &amp; 0 \\\\[1.1ex] -1  &amp; 2 &amp; 1 \\end{pmatrix}\\cdot \\begin{pmatrix} 0 &amp; 3 &amp; -3 \\\\[1.1ex] -3 &amp; 0 &amp; 3 \\\\[1.1ex] 3 &amp; -3 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"311\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Und schlie\u00dflich f\u00fchren wir die Matrixmultiplikation durch:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-552e3809229102041ddf02a78badfea0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{X=}\\begin{pmatrix} \\bm{3} &amp; \\bm{9} &amp; \\bm{-12} \\\\[1.1ex] \\bm{3} &amp; \\bm{0} &amp; \\bm{-3} \\\\[1.1ex] \\bm{-3}  &amp; \\bm{-6} &amp; \\bm{9} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"173\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Auf dieser Seite erfahren Sie, was Matrixgleichungen sind und wie Sie sie l\u00f6sen. Dar\u00fcber hinaus finden Sie Beispiele und gel\u00f6ste Aufgaben zu Gleichungen mit Matrizen. Was sind Matrixgleichungen? Matrixgleichungen sind wie normale Gleichungen, bestehen jedoch nicht aus Zahlen, sondern aus Matrizen. Zum Beispiel: Daher wird L\u00f6sung X auch eine Matrix sein. Wie Sie bereits wissen, &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/de\/beispiele-zum-losen-von-matrixgleichungen-und-geloste-ubungen-zu-2x2-und-3x3-matrizen\/\"> <span class=\"screen-reader-text\">Matrixgleichungen<\/span> Weiterlesen &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[7],"tags":[],"class_list":["post-294","post","type-post","status-publish","format-standard","hentry","category-determinante-einer-matrix"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matrixgleichungen \u2013<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/de\/beispiele-zum-losen-von-matrixgleichungen-und-geloste-ubungen-zu-2x2-und-3x3-matrizen\/\" \/>\n<meta property=\"og:locale\" content=\"de_DE\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matrixgleichungen \u2013\" \/>\n<meta property=\"og:description\" content=\"Auf dieser Seite erfahren Sie, was Matrixgleichungen sind und wie Sie sie l\u00f6sen. Dar\u00fcber hinaus finden Sie Beispiele und gel\u00f6ste Aufgaben zu Gleichungen mit Matrizen. Was sind Matrixgleichungen? Matrixgleichungen sind wie normale Gleichungen, bestehen jedoch nicht aus Zahlen, sondern aus Matrizen. Zum Beispiel: Daher wird L\u00f6sung X auch eine Matrix sein. Wie Sie bereits wissen, &hellip; Matrixgleichungen Weiterlesen &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/de\/beispiele-zum-losen-von-matrixgleichungen-und-geloste-ubungen-zu-2x2-und-3x3-matrizen\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T17:52:46+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2bc59624603d48ea9b4df50b4c052437_l3.png\" \/>\n<meta name=\"author\" content=\"Mathority Mannschaft\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Verfasst von\" \/>\n\t<meta name=\"twitter:data1\" content=\"Mathority Mannschaft\" \/>\n\t<meta name=\"twitter:label2\" content=\"Gesch\u00e4tzte Lesezeit\" \/>\n\t<meta name=\"twitter:data2\" content=\"4\u00a0Minuten\" \/>\n<script type=\"application\/ld+json\" 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