{"id":326,"date":"2023-07-06T21:48:53","date_gmt":"2023-07-06T21:48:53","guid":{"rendered":"https:\/\/mathority.org\/nl\/vermenigvuldiging-van-2x2-en-3x3-matrices-voorbeelden-en-oefeningen-stap-voor-stap-opgelost\/"},"modified":"2023-07-06T21:48:53","modified_gmt":"2023-07-06T21:48:53","slug":"vermenigvuldiging-van-2x2-en-3x3-matrices-voorbeelden-en-oefeningen-stap-voor-stap-opgelost","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/vermenigvuldiging-van-2x2-en-3x3-matrices-voorbeelden-en-oefeningen-stap-voor-stap-opgelost\/","title":{"rendered":"Matrix vermenigvuldiging"},"content":{"rendered":"<p>Op deze pagina zullen we zien hoe je <strong>matrices met de afmetingen 2\u00d72, 3\u00d73, 4\u00d74, etc. kunt vermenigvuldigen<\/strong> . We leggen de matrixvermenigvuldigingsprocedure stap voor stap uit aan de hand van een voorbeeld, daarna vind je opgeloste oefeningen zodat je ook kunt oefenen. Ten slotte leer je wanneer twee matrices niet kunnen worden vermenigvuldigd en alle eigenschappen van deze matrixbewerking.<\/p>\n<h2 class=\"wp-block-heading\"> Hoe twee matrices te vermenigvuldigen?<\/h2>\n<p> Laten we de procedure bekijken om de vermenigvuldiging van twee matrices uit te voeren met een voorbeeld: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-multiplication-matricielle-22152.webp\" alt=\"voorbeeld van het vermenigvuldigen van twee matrices met dimensie 2x2, bewerkingen met matrices\" width=\"228\" height=\"60\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Om een <strong>matrixvermenigvuldiging te berekenen,<\/strong> moeten de <strong>rijen<\/strong> van de linkermatrix worden vermenigvuldigd met de <strong>kolommen<\/strong> van de rechtermatrix.<\/p>\n<p> Dus eerst moeten we <strong>de eerste rij vermenigvuldigen met de eerste kolom.<\/strong> Om dit te doen, vermenigvuldigen we elk element in de eerste rij \u00e9\u00e9n voor \u00e9\u00e9n met elk element in de eerste kolom, en voegen we de resultaten toe. Dit alles zal dus het eerste element zijn van de eerste rij van de resulterende array. Kijk naar de werkwijze: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/comment-multiplier-des-matrices-22152.webp\" alt=\"hoe je 2x2 matrixvermenigvuldiging oplost, bewerkingen met matrices\" width=\"504\" height=\"87\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> 1 <strong>\u22c5<\/strong> 3 + 2 <strong>\u22c5<\/strong> 4 = 3 + 8 = 11. Dus: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p class=\"has-text-align-justify\"> Nu moeten we <strong>de eerste rij vermenigvuldigen met de tweede kolom<\/strong> . We herhalen daarom de procedure: we vermenigvuldigen elk element van de eerste rij \u00e9\u00e9n voor \u00e9\u00e9n met elk element van de tweede kolom, en we tellen de resultaten op. En dit alles zal het tweede element zijn van de eerste rij van de resulterende array:<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p>1 <strong>\u22c5<\/strong> 5 + 2 <strong>\u22c5<\/strong> 1 = 5 + 2 = 7. Dus: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Zodra we de eerste rij van de resulterende matrix hebben gevuld, gaan we naar de tweede rij. We vermenigvuldigen daarom <strong>de tweede rij met de eerste kolom<\/strong> door de procedure te herhalen: we vermenigvuldigen elk element van de tweede rij \u00e9\u00e9n met \u00e9\u00e9n met elk element van de eerste kolom, en tellen de resultaten op:<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p>-3 <strong>\u22c5<\/strong> 3 + 0 <strong>\u22c5<\/strong> 4 = -9 + 0 = -9. Nog: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p class=\"has-text-align-justify\"> Ten slotte vermenigvuldigen we <strong>de tweede rij met de tweede kolom<\/strong> . Altijd met dezelfde procedure: we vermenigvuldigen elk element van de tweede rij \u00e9\u00e9n voor \u00e9\u00e9n met elk element van de tweede kolom, en we tellen de resultaten op:<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p>-3 <strong>\u22c5<\/strong> 5 + 0 <strong>\u22c5<\/strong> 1 = -15 + 0 = -15. Nog:<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p>En hier eindigt de vermenigvuldiging van de twee matrices. Zoals je hebt gezien, moet je de rijen met de kolommen vermenigvuldigen, waarbij je altijd dezelfde procedure herhaalt: vermenigvuldig elk element van de rij \u00e9\u00e9n voor \u00e9\u00e9n met elk element van de kolom, en tel de resultaten bij elkaar op.<\/p>\n<h2 class=\"wp-block-heading\"> Opgeloste matrixvermenigvuldigingsoefeningen<\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Los het volgende matrixproduct op: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-produit-de-matrices-22.webp\" alt=\"oefening stap voor stap opgelost product van 2x2 matrices, bewerkingen met matrices\" width=\"172\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Het is een product van matrices van orde 2:<\/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-747926b92c1d388c1150613b0f471d7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"142\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om een matrixproduct op te lossen, moet je de rijen van de linkermatrix vermenigvuldigen met de kolommen van de rechtermatrix.<\/p>\n<p class=\"has-text-align-left has-text-align-justify\"> Dus vermenigvuldigen we eerst <strong>de eerste rij met de eerste kolom.<\/strong> Om dit te doen, vermenigvuldigen we elk element in de eerste rij \u00e9\u00e9n voor \u00e9\u00e9n met elk element in de eerste kolom, en voegen we de resultaten toe. En dit alles zal het eerste element zijn van de eerste rij van de resulterende array:<\/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-eff23eaf91738d6ffb383949e4b70856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix}  = \\begin{pmatrix} 1\\cdot 3 +2 \\cdot 1 &amp; \\\\[1.1ex] &amp; \\end{pmatrix} = \\begin{pmatrix} 5 &amp; \\\\[1.1ex] &amp; \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"370\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Laten we nu <strong>de eerste rij vermenigvuldigen met de tweede kolom<\/strong> om het tweede element van de eerste rij van de resulterende matrix te verkrijgen:<\/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-558838bcc38efc1aeeaf298d3e7151dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix}  = \\begin{pmatrix} -1 &amp; 1\\cdot (-2) +2 \\cdot 5 \\\\[1.1ex] &amp; \\end{pmatrix} = \\begin{pmatrix}5 &amp; 8 \\\\[1.1ex] &amp; \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"429\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We gaan naar de tweede rij, dus vermenigvuldigen we <strong>de tweede rij met de eerste kolom:<\/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-daab54a49cc53c320bb2965f691fd7ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix} = \\begin{pmatrix} -1 &amp; 8 \\\\[1.1ex] 3\\cdot 3 +4 \\cdot 1 &amp; \\end{pmatrix}= \\begin{pmatrix}5 &amp; 8 \\\\[1.1ex] 13 &amp; \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"396\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ten slotte vermenigvuldigen we <strong>de tweede rij met de tweede kolom<\/strong> om het laatste element van de tabel te berekenen:<\/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-a85e0d62a0db18c7712fd1b354f92bd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix}= \\begin{pmatrix} -1 &amp; 8 \\\\[1.1ex]1 &amp; 3\\cdot (-2) +4 \\cdot 5 \\end{pmatrix}=\\begin{pmatrix} 5 &amp; 8 \\\\[1.1ex] 13 &amp; 14 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"447\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het resultaat van matrixvermenigvuldiging is dus: <\/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-76f1283db0175bc1a95b0a10c8961761_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} \\bm{5} &amp; \\bm{8} \\\\[1.1ex]\\bm{13} &amp; \\bm{14} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"72\" 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\">Oefening 2<\/h3>\n<p> Zoek het resultaat van de volgende 2&#215;2 vierkante matrixvermenigvuldiging: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-multiplication-matricielle-22.webp\" alt=\"Oefening stap voor stap opgelost in 2x2 matrixvermenigvuldiging, matrixbewerkingen\" width=\"230\" height=\"70\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Het is een product van matrices met afmeting 2\u00d72.<\/p>\n<p class=\"has-text-align-left\"> Om de vermenigvuldiging op te lossen, moet je de rijen van de linkermatrix vermenigvuldigen met de kolommen van de rechtermatrix: <\/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-fc7217dab49f67df2a9d2abc561baf9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{pmatrix} 4 &amp; -1  \\\\[1.1ex] -2 &amp; 3  \\end{pmatrix} \\cdot \\begin{pmatrix} -2 &amp; 5 \\\\[1.1ex] 6 &amp; -3  \\end{pmatrix}  &amp; = \\begin{pmatrix} 4\\cdot (-2)+(-1) \\cdot 6 &amp;  4\\cdot 5+(-1) \\cdot (-3)  \\\\[1.1ex](-2)\\cdot (-2)+3 \\cdot 6 &amp; (-2)\\cdot 5+3 \\cdot (-3)\\end{pmatrix} \\\\[2ex] &amp; =\\begin{pmatrix} \\bm{-14} &amp; \\bm{23} \\\\[1.1ex]\\bm{22} &amp; \\bm{-19} \\end{pmatrix} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"528\" 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\"> Oefening 3<\/h3>\n<p> Bereken de volgende 3&#215;3 matrixvermenigvuldiging: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-multiplication-matricielle-33.webp\" alt=\"oefening opgelost stap voor stap vermenigvuldiging van 3x3 matrices, matrixbewerkingen\" width=\"277\" height=\"109\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Om een 3\u00d73-matrixvermenigvuldiging uit te voeren, moet u de rijen van de linkermatrix vermenigvuldigen met de kolommen van de rechtermatrix: <\/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-ef6ee7bb6e4ac095a9fd51a545b163b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{l} \\begin{pmatrix} 1 &amp; 2 &amp; 0 \\\\[1.1ex] 3 &amp; 2 &amp; -1 \\\\[1.1ex] 5 &amp; 1 &amp; -2  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 &amp; 0 \\\\[1.1ex] 1 &amp; 0 &amp; -2 \\\\[1.1ex] -1 &amp; 2 &amp; 1 \\end{pmatrix} = \\\\[7.5ex] =\\begin{pmatrix} 1 \\cdot 3+2 \\cdot 1+ 0 \\cdot (-1) &amp; 1 \\cdot 4+2 \\cdot 0+ 0 \\cdot 2 &amp; 1 \\cdot 0+2 \\cdot (-2)+ 0 \\cdot 1 \\\\[1.1ex] 3 \\cdot 3+2 \\cdot 1+ (-1) \\cdot (-1) &amp; 3 \\cdot 4+2 \\cdot 0+ (-1) \\cdot 2 &amp; 3 \\cdot 0+2 \\cdot (-2)+ (-1) \\cdot 1 \\\\[1.1ex] 5 \\cdot 3+1 \\cdot 1+ (-2) \\cdot (-1) &amp; 5 \\cdot 4+1 \\cdot 0+ (-2) \\cdot 2 &amp; 5 \\cdot 0+1 \\cdot (-2)+ (-2) \\cdot 1 \\end{pmatrix} = \\\\[7.5ex]  =\\begin{pmatrix} \\bm{5} &amp; \\bm{4} &amp; \\bm{-4} \\\\[1.1ex] \\bm{12} &amp; \\bm{10} &amp; \\bm{-5} \\\\[1.1ex] \\bm{18} &amp; \\bm{16} &amp; \\bm{-4} \\end{pmatrix}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"306\" width=\"643\" 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\">Oefening 4<\/h3>\n<p> gegeven de 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-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> :<\/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-27365f9993caf4fcdb747352e4ae539d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} 3 &amp; 1 &amp; -2 \\\\[1.1ex] 4 &amp; 2 &amp; -1   \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"134\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Berekenen: <\/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-307d37497055a6891b797bdb89b456e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2A\\cdot A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" 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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We zullen eerst de getransponeerde matrix berekenen van<\/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-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> om de vermenigvuldiging te doen. En om de transponeermatrix te maken, moeten we de rijen in kolommen veranderen. Dat wil zeggen dat de eerste rij van de matrix de eerste kolom van de matrix wordt en de tweede rij van de matrix de tweede kolom van de matrix. Nog:<\/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-ac4785c47f2e48e15b3d98ba426848b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^t= \\begin{pmatrix} 3 &amp; 4 \\\\[1.1ex] 1 &amp; 2  \\\\[1.1ex] -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"131\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De matrixbewerking blijft daarom:<\/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-9513fa8cc6996e18e3cf287f0210817a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2A\\cdot A^t = 2 \\begin{pmatrix} 3 &amp; 1 &amp; -2 \\\\[1.1ex] 4 &amp; 2 &amp; -1   \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 \\\\[1.1ex] 1 &amp; 2  \\\\[1.1ex] -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"291\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu kunnen we de berekeningen doen. Wij berekenen eerst<\/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-d4e94385e2fa1b091190a9ce266a8c43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\"><\/p>\n<p> (hoewel we ook eerst kunnen berekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae92cabff7a388b31fe67b559dfead7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A \\cdot A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"44\" style=\"vertical-align: 0px;\"><\/p>\n<p> ): <\/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-ae5e95f09aedac8f0861bf13fb9c78a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 2 \\cdot 3 &amp; 2 \\cdot 1 &amp; 2 \\cdot (-2) \\\\[1.1ex] 2 \\cdot 4 &amp; 2 \\cdot 2 &amp; 2 \\cdot (-1) \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 \\\\[1.1ex] 1 &amp; 2  \\\\[1.1ex] -2 &amp; -1 \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"299\" 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-24c003b8da1081d6ca494adc3356b06b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  =\\begin{pmatrix} 6 &amp; 2 &amp; -4 \\\\[1.1ex] 8 &amp; 4 &amp; -2 \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 \\\\[1.1ex] 1 &amp; 2  \\\\[1.1ex] -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"220\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte lossen we het product van matrices op: <\/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-0eb8f1817f0163a82ae39cc6c81d478e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 6 \\cdot 3 +2 \\cdot 1 + (-4) \\cdot (-2) &amp; 6 \\cdot 4 +2 \\cdot 2 + (-4) \\cdot (-1) \\\\[1.1ex] 8 \\cdot 3 +4 \\cdot 1 + (-2) \\cdot (-2) &amp; 8 \\cdot 4 +4 \\cdot 2 + (-2) \\cdot (-1) \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"438\" 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-33533be747b72497915048e486d16541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = \\begin{pmatrix} \\bm{28} &amp; \\bm{32} \\\\[1.1ex]\\bm{32} &amp; \\bm{42} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"94\" 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\">Oefening 5<\/h3>\n<p> Beschouw de volgende matrices:<\/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-6e26aec2eee6bcae0e344682d20038f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix} \\qquad B=\\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"275\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Berekenen: <\/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-78d69cf0ef5ec44cd0aacf00f4f2d613_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot B - B \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"102\" 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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Het is een bewerking die aftrekken combineert met matrixvermenigvuldigingen van orde 2:<\/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-43f79f2d970bb02caaeddec34d5ad2a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot B - B \\cdot A= \\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix}\\cdot \\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix} - \\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}  \\cdot \\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"500\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> We berekenen eerst de vermenigvuldiging aan de linkerkant: <\/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-05ff586671fb0af274884169c54e5817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2\\cdot (-1) + 4 \\cdot 3 &amp; 2\\cdot (-2) + 4 \\cdot (-3) \\\\[1.1ex] (-3)\\cdot (-1) + 5 \\cdot 3 &amp; (-3)\\cdot (-2) + 5 \\cdot (-3)  \\end{pmatrix} - \\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}  \\cdot \\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"550\" 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-43c234a2d7aa4f9dcaf3140f617480f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle= \\begin{pmatrix} 10 &amp; -16  \\\\[1.1ex] 18 &amp; -9 \\end{pmatrix} - \\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}  \\cdot \\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"308\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu lossen we de vermenigvuldiging aan de rechterkant op: <\/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-552309dd1be2f69bb72633539809283b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 10 &amp; -16  \\\\[1.1ex] 18 &amp; -9 \\end{pmatrix} - \\begin{pmatrix} -1 \\cdot 2 +(-2) \\cdot (-3) &amp;  -1 \\cdot 4 +(-2) \\cdot 5  \\\\[1.1ex]3 \\cdot 2 +(-3) \\cdot (-3) &amp;  3 \\cdot 4 +(-3) \\cdot 5  \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"449\" 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-eeac84965cc522402e869234a841ba67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle =\\begin{pmatrix} 10 &amp; -16  \\\\[1.1ex] 18 &amp; -9 \\end{pmatrix} - \\begin{pmatrix} 4 &amp;-14  \\\\[1.1ex]15 &amp; -3  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"223\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En tenslotte trekken we de matrices af: <\/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-faefbc14fc49439616b3d131243eba79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 10-4 &amp; -16 -(-14) \\\\[1.1ex] 18-15 &amp; -9-(-3) \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"214\" 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-50bac6ac99e1cf6e4b77a1a8718f9fe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle =\\begin{pmatrix} \\bm{6} &amp; \\bm{-2} \\\\[1.1ex] \\bm{3} &amp; \\bm{-6} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"90\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\">Wanneer kun je twee matrices niet vermenigvuldigen?<\/h2>\n<p> <strong>Niet alle matrices kunnen worden vermenigvuldigd.<\/strong> Om twee matrices te vermenigvuldigen, moet het aantal kolommen in de eerste matrix overeenkomen met het aantal rijen in de tweede matrix.<\/p>\n<p> De volgende vermenigvuldiging kan bijvoorbeeld niet worden uitgevoerd omdat de eerste matrix 3 kolommen heeft en de tweede matrix 2 rijen:<\/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-b8314f9238afb3676bee5c9000c02752_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{pmatrix} 1 &amp; 3 &amp; -2 \\\\[1.1ex] 4 &amp; 0 &amp; 5 \\end{pmatrix} \\cdot  \\begin{pmatrix} 2 &amp; 1  \\\\[1.1ex] 3 &amp; -1  \\end{pmatrix}  \\ \\longleftarrow \\ \\color{red} \\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"274\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Maar als we de volgorde omdraaien, kunnen ze vermenigvuldigd worden. Omdat de eerste matrix twee kolommen heeft en de tweede matrix twee rijen:<\/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-37d01cc99b578d3756312c3e6ff12cae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{pmatrix} 2 &amp; 1  \\\\[1.1ex] 3 &amp; -1  \\end{pmatrix} \\cdot \\begin{pmatrix} 1 &amp; 3 &amp; -2 \\\\[1.1ex] 4 &amp; 0 &amp; 5  \\end{pmatrix}  &amp; = \\begin{pmatrix} 2\\cdot 1 + 1 \\cdot 4 &amp; 2\\cdot 3 + 1 \\cdot 0 &amp; 2\\cdot (-2) + 1 \\cdot 5  \\\\[1.1ex] 3\\cdot 1 + (-1) \\cdot 4 &amp; 3\\cdot 3 + (-1) \\cdot 0 &amp; 3\\cdot (-2) + (-1) \\cdot 5   \\end{pmatrix} \\\\[2ex] &amp; = \\begin{pmatrix} \\bm{6} &amp; \\bm{6} &amp; \\bm{1}  \\\\[1.1ex]\\bm{-1} &amp; \\bm{9} &amp; \\bm{-11}   \\end{pmatrix}   \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"624\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Eigenschappen van matrixvermenigvuldiging<\/h2>\n<p> Dit type matrixbewerking heeft de volgende kenmerken:<\/p>\n<ul>\n<li> Matrixvermenigvuldiging is <strong><span style=\"color:#1976d2;\">associatief:<\/span><\/strong><\/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-38541ff37ecadb79ac36ffb1e19cc187_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left( A \\cdot B \\right) \\cdot C = A \\cdot \\left( B \\cdot C \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Matrixvermenigvuldiging heeft ook de <strong><span style=\"color:#1976d2;\">distributieve eigenschap:<\/span><\/strong> <\/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-1f8ca2784a9dd93cf71cd34d4d0303eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot \\left(B+C\\right) = A\\cdot B + A \\cdot C\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"216\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<ul>\n<li> Het product van matrices <strong><span style=\"color:#1976d2;\">is niet commutatief:<\/span><\/strong><\/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-67f2cce38b1aab5659a5f888daf1ff84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A \\cdot B \\neq B \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> De volgende matrixvermenigvuldiging geeft bijvoorbeeld een resultaat:<\/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-3e780b321b160ad4a612e608199a374b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{pmatrix} 1 &amp; -1  \\\\[1.1ex] 2 &amp; 3  \\end{pmatrix} \\cdot \\begin{pmatrix} -2 &amp; 5  \\\\[1.1ex] 0 &amp; 1   \\end{pmatrix}  &amp; = \\begin{pmatrix} 1\\cdot (-2) + (-1) \\cdot 0 &amp; 1\\cdot 5 + (-1) \\cdot 1   \\\\[1.1ex] 2\\cdot (-2) + 3 \\cdot 0 &amp;  2\\cdot 5 + 3 \\cdot 1    \\end{pmatrix} \\\\[2ex] &amp; = \\begin{pmatrix} \\bm{-2} &amp; \\bm{4} \\\\[1.1ex] \\bm{-4} &amp;  \\bm{13} \\end{pmatrix}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"472\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Maar het resultaat van het product is anders als we de volgorde van vermenigvuldiging van de matrices omkeren:<\/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-177d78a209e5d9e18828617e4913176d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned}\\begin{pmatrix} -2 &amp; 5  \\\\[1.1ex] 0 &amp; 1   \\end{pmatrix} \\cdot  \\begin{pmatrix} 1 &amp; -1  \\\\[1.1ex] 2 &amp; 3  \\end{pmatrix} &amp; = \\begin{pmatrix} -2 \\cdot 1 + 5\\cdot 2 &amp;  -2 \\cdot (-1) + 5\\cdot 3  \\\\[1.1ex] 0 \\cdot 1 + 1\\cdot 2 &amp;  0 \\cdot (-1) + 1\\cdot 3   \\end{pmatrix} \\\\[2ex] &amp; = \\begin{pmatrix} \\bm{8} &amp;  \\bm{17}  \\\\[1.1ex] \\bm{2} &amp;  \\bm{3} \\end{pmatrix}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"445\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Bovendien resulteert elke matrix vermenigvuldigd met de identiteitsmatrix in dezelfde matrix. Dit wordt <strong><span style=\"color:#1976d2;\">de multiplicatieve identiteitseigenschap genoemd:<\/span><\/strong><\/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-7ab05972282922f1e10f75a50e636887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A \\cdot I=A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"72\" 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-2c32986a7c34108a47500a4f0ec2967b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle I \\cdot A=A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"72\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Bijvoorbeeld:<\/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-9c1e72173419eb76554256cf6ccd0d2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 2 &amp; 7  \\\\[1.1ex] -6 &amp; 5  \\end{pmatrix} \\cdot \\begin{pmatrix} 1 &amp; 0  \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix} = \\begin{pmatrix} \\bm{2} &amp; \\bm{7}  \\\\[1.1ex] \\bm{-6} &amp; \\bm{5}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"242\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Ten slotte is, zoals je misschien al vermoedt, elke matrix vermenigvuldigd met de nulmatrix gelijk aan de nulmatrix. Dit wordt <strong><span style=\"color:#1976d2;\">de multiplicatieve eigenschap van nul genoemd:<\/span><\/strong><\/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-cf700c38f25e0c3bdf1c46851341a815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A \\cdot 0=0\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" 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-ac3340bc96ba3df60f6ddeb6bbd3b4b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0\\cdot A=0\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Bijvoorbeeld:<\/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-3152d82054a80d61d548e969290aea4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 6 &amp; -4  \\\\[1.1ex] 3 &amp; 8  \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; 0  \\\\[1.1ex] 0 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} \\bm{0} &amp; \\bm{0}  \\\\[1.1ex] \\bm{0} &amp; \\bm{0}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"228\" style=\"vertical-align: 0px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Op deze pagina zullen we zien hoe je matrices met de afmetingen 2\u00d72, 3\u00d73, 4\u00d74, etc. kunt vermenigvuldigen . We leggen de matrixvermenigvuldigingsprocedure stap voor stap uit aan de hand van een voorbeeld, daarna vind je opgeloste oefeningen zodat je ook kunt oefenen. Ten slotte leer je wanneer twee matrices niet kunnen worden vermenigvuldigd en &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/vermenigvuldiging-van-2x2-en-3x3-matrices-voorbeelden-en-oefeningen-stap-voor-stap-opgelost\/\"> <span class=\"screen-reader-text\">Matrix vermenigvuldiging<\/span> Lees meer &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":[52],"tags":[],"class_list":["post-326","post","type-post","status-publish","format-standard","hentry","category-schilderijen"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matrixvermenigvuldiging - Mathority<\/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\/nl\/vermenigvuldiging-van-2x2-en-3x3-matrices-voorbeelden-en-oefeningen-stap-voor-stap-opgelost\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matrixvermenigvuldiging - Mathority\" \/>\n<meta property=\"og:description\" content=\"Op deze pagina zullen we zien hoe je matrices met de afmetingen 2\u00d72, 3\u00d73, 4\u00d74, etc. kunt vermenigvuldigen . 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