{"id":327,"date":"2023-07-06T21:19:23","date_gmt":"2023-07-06T21:19:23","guid":{"rendered":"https:\/\/mathority.org\/nl\/machten-van-2x2-en-3x3-matrices-voorbeelden-en-opgeloste-oefeningen\/"},"modified":"2023-07-06T21:19:23","modified_gmt":"2023-07-06T21:19:23","slug":"machten-van-2x2-en-3x3-matrices-voorbeelden-en-opgeloste-oefeningen","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/machten-van-2x2-en-3x3-matrices-voorbeelden-en-opgeloste-oefeningen\/","title":{"rendered":"Matrix-krachten"},"content":{"rendered":"<p>Op deze pagina zullen we zien hoe je <strong>machten van matrices kunt gebruiken.<\/strong> Je vindt er ook voorbeelden en stap voor stap opgeloste oefeningen van machten van matrices die je zullen helpen het perfect te begrijpen. Ook leer je wat de n-de macht van een matrix is en hoe je deze kunt vinden.<\/p>\n<h2 class=\"wp-block-heading\"> Hoe wordt de kracht van een matrix berekend? <\/h2>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\"> Om de <strong>macht van een matrix<\/strong> te berekenen, moet je de matrix zo vaak met zichzelf vermenigvuldigen als de exponent aangeeft. 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-3e77b01db3eabfb211a806dcae2fc5c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^4 = A \\cdot A \\cdot A \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<p> Om de kracht van een matrix te krijgen, moet je daarom weten hoe je <a href=\"https:\/\/mathority.org\/nl\/vermenigvuldiging-van-2x2-en-3x3-matrices-voorbeelden-en-oefeningen-stap-voor-stap-opgelost\/\">matrixvermenigvuldiging<\/a> oplost. Anders kun je geen machtsmatrix berekenen.<\/p>\n<h3 class=\"wp-block-heading\"> Voorbeeld van het berekenen van de kracht van een matrix: <\/h3>\n<figure class=\"wp-block-image aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemples-de-puissances-de-matrices-22.webp\" alt=\"voorbeelden van machten van 2x2 matrices\" width=\"560\" height=\"471\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Daarom wordt de kracht van een vierkante matrix berekend door de matrix met zichzelf te vermenigvuldigen. Op dezelfde manier is een kubusvormige matrix gelijk aan de vierkante matrix van de matrix zelf. Op dezelfde manier moet, om de macht te vinden van een matrix verhoogd tot vier, de matrix verhoogd tot drie worden vermenigvuldigd met de matrix zelf. Enzovoort.<\/p>\n<p> Er is een belangrijke eigenschap van matrixmacht die u moet kennen: <strong>de macht van een matrix kan alleen worden berekend als deze vierkant is<\/strong> , dat wil zeggen als deze hetzelfde aantal rijen als kolommen heeft.<\/p>\n<h2 class=\"wp-block-heading\"> Wat is de macht n van een matrix?<\/h2>\n<p> De <strong>n-de macht van een matrix<\/strong> is een uitdrukking waarmee we eenvoudig elke macht van een matrix kunnen berekenen.<\/p>\n<p> Vaak volgen de krachten van matrices een <strong>patroon<\/strong> . Als we de reeks die ze volgen kunnen ontcijferen, kunnen we dus elke macht berekenen zonder alle vermenigvuldigingen te hoeven doen.<\/p>\n<p> Dit betekent dat we een formule kunnen vinden die ons de n-de macht van een matrix geeft zonder alle machten te hoeven berekenen. <\/p>\n<div style=\"background-color:#fffde7;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\"> <strong>Tips<\/strong> voor het ontdekken van het patroon gevolgd door de krachten:<\/p>\n<ul style=\"color:#1976d2; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#000000;font-weight: normal;\">De <strong>pariteit van de exponent<\/strong> . Het kan zijn dat zelfs krachten de ene kant op gaan en oneven krachten de andere kant op.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\"><strong>Variatie van tekens.<\/strong> Het kan bijvoorbeeld zijn dat elementen van even machten positief zijn en elementen van oneven machten negatief, of omgekeerd.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\"><strong>Herhaling:<\/strong> of dezelfde matrix elk bepaald aantal machten wordt herhaald of niet.<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">We moeten ook kijken of er een <strong>relatie<\/strong> bestaat tussen de exponent en de elementen van de matrix.<\/span> <\/li>\n<\/ul>\n<\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Voorbeeld van het berekenen van de macht n van een matrix:<\/h3>\n<ul>\n<li> Zijn\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> bereken de volgende matrix<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-34564dd93ab535fd300f9ac993829376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^n\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: 0px;\"><\/p>\n<p> En<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-52b77e64505e02204c8e501aea82c251_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^{100}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"34\" style=\"vertical-align: 0px;\"><\/p>\n<p> .<\/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-60016ce1c6799c93007526681fbf4894_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A = \\begin{pmatrix} 1 &amp; 1 \\\\[1.1ex] 1 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> We zullen eerst een aantal machten van de matrix 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-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 te proberen het patroon te raden dat door de krachten wordt gevolgd. Dus wij berekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8b49aeb7162689d03dd9f9470a2ae1a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" 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-07e0009cbaebcb5501371dd9f6795f4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" 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-2ccb300f7879fa598883dafb53bf7a5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^4\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> En <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f2ce79bf092ea6898cbcbc086729ba93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^5:\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"30\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<figure class=\"wp-block-image aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-pas-a-pas-des-puissances-des-matrices-22.webp\" alt=\"oefening stap voor stap opgelost van de machten van 2x2 matrices\" width=\"409\" height=\"361\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Bij het berekenen tot<\/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-0d1e5d53cda856213bbb6b5796706dd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^5\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> , we zien dat de krachten van de matrix<\/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> Ze volgen een patroon: voor elke toename van het vermogen wordt het resultaat vermenigvuldigd met 2. Daarom <strong>zijn alle matrices machten van 2:<\/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-4ec7ee835cf9eda6a4f9d497e8baff79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2= \\begin{pmatrix} 2 &amp; 2 \\\\[1.1ex] 2 &amp; 2 \\end{pmatrix} =\\begin{pmatrix} 2^1 &amp; 2^1 \\\\[1.1ex] 2^1 &amp; 2^1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"204\" 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-69c6ff0f4de92192584dadc4719167c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3= \\begin{pmatrix} 4 &amp; 4 \\\\[1.1ex] 4 &amp; 4 \\end{pmatrix}=\\begin{pmatrix} 2^2 &amp; 2^2 \\\\[1.1ex] 2^2 &amp; 2^2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"204\" 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-f724a50b220b3026d53e40ee17870359_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4= \\begin{pmatrix} 8 &amp; 8 \\\\[1.1ex] 8 &amp; 8 \\end{pmatrix}=\\begin{pmatrix} 2^3 &amp; 2^3 \\\\[1.1ex] 2^3 &amp; 2^3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"204\" 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-f5f08f7cc00465a6a098ce7d752aa66f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5= \\begin{pmatrix} 16 &amp; 16 \\\\[1.1ex] 16 &amp; 16 \\end{pmatrix}=\\begin{pmatrix} 2^4 &amp; 2^4 \\\\[1.1ex] 2^4 &amp; 2^4 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"221\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> We kunnen daarom de formule voor de <strong>n-de macht<\/strong> van de matrix afleiden <\/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-944477c7f7578892a57aa3b7c7dd8268_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A:\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<figure class=\"wp-block-image aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/nieme-puissance-dune-matrice.webp\" alt=\"nde macht van een 2x2 matrix\" width=\"201\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> En met deze formule kunnen we 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-560982f344534dee89eb7afbf6be520e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^{100}:\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"44\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<figure class=\"wp-block-image aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-de-puissance-resolu-dune-matrice.webp\" alt=\"oefening stap voor stap opgelost macht van een 2x2 matrix\" width=\"187\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"> Matrixmachtsproblemen opgelost<\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Beschouw de volgende matrix van afmeting 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-cdf81cf9fb956a144c7bda96a84ec7db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] -1 &amp; 1  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"109\" 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-2589110bbf0eae4fa44ef48ab7b0f416_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" 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\"> Om de macht van een matrix te berekenen, moet je de matrix \u00e9\u00e9n voor \u00e9\u00e9n vermenigvuldigen. Daarom berekenen wij 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-d7581934ef6136b2b48380f1a53c7809_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2 :\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"30\" 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-24916b0b0e4431b0a2ee2b09875dc903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2= A \\cdot A = \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] -1 &amp; 1 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] -1 &amp; 1 \\end{pmatrix} = \\begin{pmatrix} -1 &amp; 4 \\\\[1.1ex] -2 &amp;  -1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"381\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nu gaan we 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-fecf45671ed5e89f1f756fd265fcf13b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3 :\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"30\" 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-57f79bd420c0044c84a64b431035b8ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3= A^2 \\cdot A = \\begin{pmatrix} -1 &amp; 4 \\\\[1.1ex] -2 &amp;  -1 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] -1 &amp; 1 \\end{pmatrix} =\\begin{pmatrix} -5 &amp; 2 \\\\[1.1ex] -1 &amp;  -5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"403\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En tot slot gaan we rekenen <\/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-f95589f39821fada84cb5b3d4ba91a46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4 :\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"30\" 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-bbc2ad8229ee141b323c9bbcc9df00fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4= A^3 \\cdot A = \\begin{pmatrix} -5 &amp; 2 \\\\[1.1ex] -1 &amp;  -5 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] -1 &amp; 1 \\end{pmatrix} = \\begin{pmatrix} \\bm{-7} &amp; \\bm{-8} \\\\[1.1ex] \\bm{4} &amp;  \\bm{-7} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"403\" 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> Beschouw de volgende matrix 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-33db03560b5c28f45eef9aa293484603_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" 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-6f350af4394f9224a8a2d726ed6ed0aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{35}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" 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-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6f350af4394f9224a8a2d726ed6ed0aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{35}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<p> is een te grote macht om met de hand te berekenen, dus de matrixmachten moeten een patroon volgen. Dus laten we berekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0678e990fe5d8fe1614d53eb51816f13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> om te proberen de volgorde te begrijpen die ze volgen: <\/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-cb9646cc984d754d2a618e6223e93cd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2= A \\cdot A = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3 \\end{pmatrix} \\cdot \\begin{pmatrix}1 &amp; 0 \\\\[1.1ex] 0 &amp; 3 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 9 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"326\" 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-22fdee28399b9115de98a214ba0c8473_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3= A^2 \\cdot A = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 9 \\end{pmatrix} \\cdot \\begin{pmatrix}1 &amp; 0 \\\\[1.1ex] 0 &amp; 3 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 27 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"343\" 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-1a085a2338ce1e74885ca04bbd0011a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4= A^3 \\cdot A = \\begin{pmatrix}1 &amp; 0 \\\\[1.1ex] 0 &amp; 27 \\end{pmatrix} \\cdot \\begin{pmatrix}1 &amp; 0 \\\\[1.1ex] 0 &amp; 3 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 81 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"351\" 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-3dc357146829da8323a0755fa16a8ca8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5= A^4 \\cdot A = \\begin{pmatrix}1 &amp; 0 \\\\[1.1ex] 0 &amp; 81 \\end{pmatrix} \\cdot \\begin{pmatrix}1 &amp; 0 \\\\[1.1ex] 0 &amp; 3 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 243 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"360\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zo kunnen we het patroon zien dat de machten volgen: bij elke macht blijven alle getallen hetzelfde, behalve het element in de tweede kolom van de tweede rij, dat wordt vermenigvuldigd met 3. Daarom <strong>blijven alle getallen altijd hetzelfde. en het laatste element is een macht van 3:<\/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-a0bfa34768808832e0fd5d3f730eb27b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3 \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3^1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"188\" 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-f6e007f5ad5d38fd887d39f00bd2b9fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 9 \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3^2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"196\" 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-585d8a00f418b50f60b4f95d87c5839c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 27 \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3^3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"205\" 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-dec6b9db4b59d9759adf85cee442cca3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 81 \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3^4 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"205\" 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-f7244b46950df4d9107cbdb7ad004e17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 243 \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3^5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"214\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus de formule voor <strong>de n-de macht van de matrix<\/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-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> Oosten:<\/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-beec2f1ed3e47902de0f25fe1901e294_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^n=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 3^n\\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\"> En met deze formule kunnen we 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-c4057ee894404b505d020a186733732e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{35}:\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"37\" 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-aa3261646ca7bfa41f8ad46331a0af4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\bm{A^{35}=}\\begin{pmatrix} \\bm{1} &amp; \\bm{0} \\\\[1.1ex] \\bm{0} &amp; \\bm{3^{35}}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"122\" 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 3<\/h3>\n<p> Beschouw de volgende 3\u00d73-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-f11fe8a7dcd1e308faa0af24eee3f362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; \\frac{1}{5}  &amp; \\frac{1}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"126\" 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-a99c928415cd39eb81240e79778e41df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{100}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"34\" 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-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a99c928415cd39eb81240e79778e41df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{100}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"34\" style=\"vertical-align: 0px;\"><\/p>\n<p> is een te grote macht om met de hand te berekenen, dus de matrixmachten moeten een patroon volgen. Dus laten we berekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0678e990fe5d8fe1614d53eb51816f13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> om te proberen de volgorde te begrijpen die ze volgen: <\/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-acb15d7f461d11e3668bc0b96a1fdc06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2= A \\cdot A = \\begin{pmatrix} 1 &amp; \\frac{1}{5}  &amp; \\frac{1}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix} \\cdot \\begin{pmatrix}1 &amp; \\frac{1}{5}  &amp; \\frac{1}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix} =  \\begin{pmatrix} 1 &amp; \\frac{2}{5}   &amp; \\frac{2}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"421\" 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-f416625ded948830fa80799249c12608_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3= A^2 \\cdot A = \\begin{pmatrix} 1 &amp; \\frac{2}{5}   &amp; \\frac{2}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1\\end{pmatrix} \\cdot \\begin{pmatrix}1 &amp; \\frac{1}{5}  &amp; \\frac{1}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; \\frac{3}{5}   &amp; \\frac{3}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"429\" 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-a76fd60051b157f06c2a731ff575d1e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4= A^3 \\cdot A = \\begin{pmatrix} 1 &amp; \\frac{3}{5}   &amp; \\frac{3}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1\\end{pmatrix} \\cdot \\begin{pmatrix}1 &amp; \\frac{1}{5}  &amp; \\frac{1}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix} =  \\begin{pmatrix} 1 &amp; \\frac{4}{5}   &amp; \\frac{4}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"429\" 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-3409c7b8d82ffd21cc084a12405fce74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5= A^4 \\cdot A = \\begin{pmatrix} 1 &amp; \\frac{4}{5}   &amp; \\frac{4}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1\\end{pmatrix} \\cdot \\begin{pmatrix}1 &amp; \\frac{1}{5}  &amp; \\frac{1}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix} =  \\begin{pmatrix} 1 &amp; \\frac{5}{5}   &amp; \\frac{5}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"429\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Zo kunnen we het patroon zien dat de machten volgen: bij elke macht blijven alle getallen hetzelfde, behalve breuken, die <strong>in de teller met \u00e9\u00e9n toenemen:<\/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-86c72aa2b21e7a68bbebfe7af5daa420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; \\frac{1}{5}   &amp; \\frac{1}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"126\" 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-ce805455e49bf018f8f22588391ac44c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2= \\begin{pmatrix} 1 &amp; \\frac{2}{5}   &amp; \\frac{2}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"134\" 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-bd5468ece9001274493687f3786b0af3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3= \\begin{pmatrix} 1 &amp; \\frac{3}{5}   &amp; \\frac{3}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"134\" 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-07fd0e03c0163b58fffbe0235009fd8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4= \\begin{pmatrix} 1 &amp; \\frac{4}{5}   &amp; \\frac{4}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"134\" 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-5ea88723757d1f2d8d6de1ac2d3843c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5= \\begin{pmatrix} 1 &amp; \\frac{5}{5}   &amp; \\frac{5}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"134\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dus de formule voor <strong>de macht van de <strong>n-<\/strong> de matrix<\/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-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> Oosten:<\/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-56308ff348d67ba1aba5816d85e9ee1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^n= \\begin{pmatrix} 1 &amp; \\frac{n}{5}   &amp; \\frac{n}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"138\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En met deze formule kunnen we 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-d22628ae2f8152f9817b84fa09c97d6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{100}:\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" 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-5352f021f5ab30e999c57f978ff55ad6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{100}=   \\begin{pmatrix} 1 &amp; \\frac{100}{5}   &amp; \\frac{100}{5} \\\\[1.1ex] 0 &amp; 1  &amp; 0 \\\\[1.1ex] 0 &amp; 0  &amp; 1 \\end{pmatrix}= \\begin{pmatrix} \\bm{1} &amp; \\bm{20}   &amp; \\bm{20} \\\\[1.1ex] \\bm{0} &amp; \\bm{1}  &amp; \\bm{0} \\\\[1.1ex] \\bm{0} &amp; \\bm{0}  &amp; \\bm{1} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"307\" 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 4<\/h3>\n<p> Beschouw de volgende matrix van maat 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-4609248b534d656aa9495b58f42e343f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 0 &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> 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-4f8edde6fcaa57b102140f3d4437f95b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{201}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"33\" 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-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f8edde6fcaa57b102140f3d4437f95b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{201}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"33\" style=\"vertical-align: 0px;\"><\/p>\n<p> is een te grote macht om met de hand te berekenen, dus de matrixmachten moeten een patroon volgen. In dit geval is het noodzakelijk om te berekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8f4a7b26a48a1e57dc08ef4c8c662af6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{8}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> om de volgorde te kennen die ze volgen: <\/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-c9a1fb4cf8bb75cf02d76a26054e6bfa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2= A \\cdot A = \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} -1 &amp; 0 \\\\[1.1ex] 0 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"381\" 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-110c4b30c78811cafdd4234e128ed414_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3= A^2 \\cdot A = \\begin{pmatrix} -1 &amp; 0 \\\\[1.1ex] 0 &amp; -1 \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 0 &amp; 1 \\\\[1.1ex] -1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"389\" 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-2b1976bbdf3c1daa9d75497efc07975c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4= A^3 \\cdot A = \\begin{pmatrix}0 &amp; 1 \\\\[1.1ex] -1 &amp; 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix} = \\bm{I}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"398\" 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-e0266d832a2fc0a04c9f6582dc231d57_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5= A^4 \\cdot A = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1\\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"361\" 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-21dea9844b7bfdb990bbb2bc955c866e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^6= A^5 \\cdot A = \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} -1 &amp; 0 \\\\[1.1ex] 0 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"389\" 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-788e75a71c1dfe4a60f0e52960715efe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^7= A^6 \\cdot A = \\begin{pmatrix} -1 &amp; 0 \\\\[1.1ex] 0 &amp; -1 \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 0 &amp; 1 \\\\[1.1ex] -1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"389\" 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-4947286a163847383e3735a508b0037d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^8= A^7 \\cdot A = \\begin{pmatrix}0 &amp; 1 \\\\[1.1ex] -1 &amp; 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix} = \\bm{I}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"398\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Met deze berekeningen kunnen we zien dat we voor elke vier machten de identiteitsmatrix krijgen. Dat wil zeggen dat het ons als resultaat de identiteitsmatrix van de machten zal opleveren<\/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-2589110bbf0eae4fa44ef48ab7b0f416_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" 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-b6df3f4d3068241a434e489e7f1d655e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^8\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" 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-d390d2dcb2acd63a2b3af76fa1451d29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{12}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" 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-26e32d520eee6a2f5c39f1d6de0c9ffc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{16}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<p> ,\u2026 Dus om uit te rekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f8edde6fcaa57b102140f3d4437f95b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{201}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"33\" style=\"vertical-align: 0px;\"><\/p>\n<p> we moeten 201 ontbinden in veelvouden van 4: <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-etape-par-etape-puissance-dune-matrice.webp\" alt=\"oefening stap voor stap opgelost van de machten van 2x2 matrices en macht n\" class=\"wp-image-327\" width=\"416\" height=\"160\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3c705236856598d218f071b1ca9a370d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 201= 4 \\cdot 50 +1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"119\" style=\"vertical-align: -2px;\"><\/p>\n<p> ,Nog,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-01a8a8f62467b5a911593c44559f2dc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^{201}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"33\" style=\"vertical-align: 0px;\"><\/p>\n<p> het zal 50 keer zijn<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1483b12f3e81520e751acccec37f9c21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> en eenmaal<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3937de4ff8cc137d41d4ac1bbccf561c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{1}:\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"30\" 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-0e169084d9ac06e6c2895a2b1f4be3f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{201}=\\left(A^4 \\right)^{50} \\cdot A^1\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"142\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En hoe weten wij dat<\/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-2589110bbf0eae4fa44ef48ab7b0f416_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> is de identiteitsmatrix <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-867357beec26a26d9d9b4af01b8086e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle I :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" 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-e3f630d4fa8da50f18be6835617a6982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4 =I\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" 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-29c53c0280332f200d37936b211faf39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{201}=\\left(A^4 \\right)^{50} \\cdot A^1 = I^{50}\\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"217\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Bovendien geeft de identiteitsmatrix, verhoogd tot een willekeurig getal, de identiteitsmatrix. 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-f0748e850cbae2f5a2d9eb797e27641b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{201}= I^{50}\\cdot A = I \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"167\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En ten slotte geeft elke matrix vermenigvuldigd met de identiteitsmatrix dezelfde matrix. 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-c88ebfbbdcc01a0cbdcf840aba32313e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{201}= I \\cdot A = A\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"130\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Waarvoor<\/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-01a8a8f62467b5a911593c44559f2dc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^{201}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"33\" style=\"vertical-align: 0px;\"><\/p>\n<p> is gelijk aan <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-944477c7f7578892a57aa3b7c7dd8268_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A:\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" 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-1214abe876a5aede8fbbce79009d5dbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{201}= A =\\begin{pmatrix} \\bm{0} &amp; \\bm{-1} \\\\[1.1ex] \\bm{1} &amp; \\bm{0} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"167\" 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 matrix van orde 3:<\/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-b8f3ba8b2d15b622f99774be05aa2620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"164\" 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-bd886e003dc8d850cca00cfe4d00ed4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" 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\"> Bereken uiteraard de kracht van 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-bd886e003dc8d850cca00cfe4d00ed4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dit is een te grote berekening om met de hand uit te voeren, dus de matrixmachten moeten een patroon volgen. In dit geval is het noodzakelijk om te berekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b9dcf97a16a30b4167b19a2313ee060c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> om de volgorde te kennen die ze volgen: <\/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-4032b55d68a5615911a5b7c997b05e6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^2= A \\cdot A = \\begin{pmatrix}3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 3 &amp; 3 &amp; 1 \\\\[1.1ex] -2 &amp; -2 &amp; -1 \\\\[1.1ex] 0 &amp; 1 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"534\" 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-8b5deef2a7728c5e82e1a1dafb1a939c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3= A^2 \\cdot A = \\begin{pmatrix}3 &amp; 3 &amp; 1 \\\\[1.1ex] -2 &amp; -2 &amp; -1 \\\\[1.1ex] 0 &amp; 1 &amp; -1\\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 1 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"500\" 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-f62e856d037138b2ead39b17ccebf96d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^4= A^3 \\cdot A = \\begin{pmatrix}1 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 1 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 1 \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"500\" 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-854da5c09b6662da46acb790afb6d01a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^5= A^4 \\cdot A = \\begin{pmatrix}3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 3 &amp; 3 &amp; 1 \\\\[1.1ex] -2 &amp; -2 &amp; -1 \\\\[1.1ex] 0 &amp; 1 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"541\" 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-c9f804a1c129e18d105fb92254c971fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^6= A^5 \\cdot A = \\begin{pmatrix}3 &amp; 3 &amp; 1 \\\\[1.1ex] -2 &amp; -2 &amp; -1 \\\\[1.1ex] 0 &amp; 1 &amp; -1\\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex] -2 &amp; -3 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 1 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"500\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Met deze berekeningen kunnen we zien dat we voor elke 3 machten de identiteitsmatrix verkrijgen. Dat wil zeggen dat het ons als resultaat de identiteitsmatrix van de machten zal opleveren<\/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-ca00633b1d21d63a177e78aed3846413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" 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-33a1b80dd4db27f09aa071e4b8bf01a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^6\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" 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-4c2f4eb36ca05968a81ef76d76e9275c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{9}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" 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-d390d2dcb2acd63a2b3af76fa1451d29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{12}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<p> ,\u2026Dus dat is te berekenen<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd886e003dc8d850cca00cfe4d00ed4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<p> We moeten 62 ontbinden in veelvouden van 3: <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-des-puissances-des-matrices-33.webp\" alt=\"oefening stap voor stap opgelost van een macht van een 3x3 matrix, nde macht\" class=\"wp-image-339\" width=\"394\" height=\"160\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f1ebd498146526b26797fc73174c6bef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 62= 3 \\cdot 20 +2\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"110\" style=\"vertical-align: -2px;\"><\/p>\n<p> ,Nog,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd886e003dc8d850cca00cfe4d00ed4a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<p> het zal 20 keer zijn<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6129a88e40a1a7fa3b922c8ef6ec57cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> en eenmaal<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-490432e07ef01473684f6a975567a3d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{2}:\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"30\" 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-db1749b0c96e2613326aa9bac2cbf651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}=\\left(A^3 \\right)^{20} \\cdot A^2\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"136\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En hoe weten wij dat<\/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-ca00633b1d21d63a177e78aed3846413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> is de identiteitsmatrix <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-867357beec26a26d9d9b4af01b8086e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle I :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" 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-e4af75581d64edceeaa20edefbde7d8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^3 =I\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" 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-c885875cfd8f37ead41f1b9cae94a3f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}=\\left(A^3 \\right)^{20} \\cdot A^2 = I^{20}\\cdot A^2\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"217\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Bovendien geeft de identiteitsmatrix, verhoogd tot een willekeurig getal, de identiteitsmatrix. 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-a3175b230605c5218a3fc03c53cbd14b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}= I^{20}\\cdot A^2 = I \\cdot A^2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"175\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ten slotte geeft elke matrix vermenigvuldigd met de identiteitsmatrix dezelfde 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-269a862d24453f1dff22c4599b6fa775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}= I \\cdot A^2 = A^2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"138\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Waarvoor<\/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-23af6c06fb07a3267b3401415f6c0449_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^{62}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\"><\/p>\n<p> gelijk zal zijn aan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e1844da717e117a743161ee5e453ae3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: 0px;\"><\/p>\n<p> , waarvoor we eerder het resultaat hebben berekend:<\/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-3f95e17aacde501ca1c28dbf14324f0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{62}= A^2=\\begin{pmatrix} \\bm{3} &amp; \\bm{3} &amp; \\bm{1} \\\\[1.1ex] \\bm{-2} &amp; \\bm{-2} &amp; \\bm{-1} \\\\[1.1ex] \\bm{0} &amp; \\bm{1} &amp; \\bm{-1} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"223\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Als deze oefeningen over de machten van vierkante matrices nuttig voor je waren, kun je ook opgeloste stapsgewijze oefeningen vinden over het optellen en <a href=\"https:\/\/mathority.org\/nl\/optellen-aftrekken-van-matrices-2x2-3x3-voorbeelden-opgeloste-oefeningen\/\">aftrekken van matrices<\/a> , een van de meest gebruikte bewerkingen met matrices.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Op deze pagina zullen we zien hoe je machten van matrices kunt gebruiken. Je vindt er ook voorbeelden en stap voor stap opgeloste oefeningen van machten van matrices die je zullen helpen het perfect te begrijpen. Ook leer je wat de n-de macht van een matrix is en hoe je deze kunt vinden. Hoe wordt &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/machten-van-2x2-en-3x3-matrices-voorbeelden-en-opgeloste-oefeningen\/\"> <span class=\"screen-reader-text\">Matrix-krachten<\/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":[39],"tags":[],"class_list":["post-327","post","type-post","status-publish","format-standard","hentry","category-determinant-van-een-matrix"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matrixkrachten -<\/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\/machten-van-2x2-en-3x3-matrices-voorbeelden-en-opgeloste-oefeningen\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matrixkrachten -\" \/>\n<meta property=\"og:description\" content=\"Op deze pagina zullen we zien hoe je machten van matrices kunt gebruiken. Je vindt er ook voorbeelden en stap voor stap opgeloste oefeningen van machten van matrices die je zullen helpen het perfect te begrijpen. Ook leer je wat de n-de macht van een matrix is en hoe je deze kunt vinden. 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