{"id":282,"date":"2023-07-06T21:19:23","date_gmt":"2023-07-06T21:19:23","guid":{"rendered":"https:\/\/mathority.org\/pt\/potencias-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos\/"},"modified":"2023-07-06T21:19:23","modified_gmt":"2023-07-06T21:19:23","slug":"potencias-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/potencias-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos\/","title":{"rendered":"Poderes da matriz"},"content":{"rendered":"<p>Nesta p\u00e1gina veremos como fazer <strong>pot\u00eancias de matrizes.<\/strong> Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos passo a passo de pot\u00eancias de matrizes que o ajudar\u00e3o a entend\u00ea-lo perfeitamente. Voc\u00ea tamb\u00e9m aprender\u00e1 o que \u00e9 a en\u00e9sima pot\u00eancia de uma matriz e como encontr\u00e1-la.<\/p>\n<h2 class=\"wp-block-heading\"> Como \u00e9 calculada a pot\u00eancia de uma matriz? <\/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\"> Para calcular a <strong>pot\u00eancia de uma matriz<\/strong> , voc\u00ea deve multiplicar a matriz por ela mesma quantas vezes o expoente indicar. Por exemplo:<\/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> Portanto, para obter a pot\u00eancia de uma matriz, voc\u00ea precisa saber como resolver <a href=\"https:\/\/mathority.org\/pt\/multiplicacao-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos-passo-a-passo\/\">a multiplica\u00e7\u00e3o de matrizes<\/a> . Caso contr\u00e1rio, voc\u00ea n\u00e3o poder\u00e1 calcular uma matriz de pot\u00eancia.<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo de c\u00e1lculo da pot\u00eancia de uma matriz: <\/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=\"exemplos de pot\u00eancias de matrizes 2x2\" width=\"560\" height=\"471\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Portanto, a pot\u00eancia de uma matriz quadrada \u00e9 calculada multiplicando a matriz por ela mesma. Da mesma forma, uma matriz c\u00fabica \u00e9 igual \u00e0 matriz quadrada da pr\u00f3pria matriz. Da mesma forma, para encontrar a pot\u00eancia de uma matriz elevada a quatro, a matriz elevada a tr\u00eas deve ser multiplicada pela pr\u00f3pria matriz. E assim por diante.<\/p>\n<p> Existe uma propriedade importante da pot\u00eancia da matriz que voc\u00ea deve conhecer: <strong>a pot\u00eancia de uma matriz s\u00f3 pode ser calculada quando ela \u00e9 quadrada<\/strong> , ou seja, quando possui o mesmo n\u00famero de linhas que de colunas.<\/p>\n<h2 class=\"wp-block-heading\"> Qual \u00e9 a pot\u00eancia n de uma matriz?<\/h2>\n<p> A <strong>en\u00e9sima pot\u00eancia de uma matriz<\/strong> \u00e9 uma express\u00e3o que nos permite calcular facilmente qualquer pot\u00eancia de uma matriz.<\/p>\n<p> Muitas vezes as pot\u00eancias das matrizes seguem um <strong>padr\u00e3o<\/strong> . Portanto, se conseguirmos decifrar a sequ\u00eancia que seguem, poderemos calcular qualquer pot\u00eancia sem ter que fazer todas as multiplica\u00e7\u00f5es.<\/p>\n<p> Isso significa que podemos encontrar uma f\u00f3rmula que nos d\u00ea a en\u00e9sima pot\u00eancia de uma matriz sem precisar calcular todas as pot\u00eancias. <\/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>Dicas<\/strong> para descobrir o padr\u00e3o seguido pelos poderes:<\/p>\n<ul style=\"color:#1976d2; font-weight: bold;\">\n<li style=\"margin-bottom:16px\"> <span style=\"color:#000000;font-weight: normal;\">A <strong>paridade do expoente<\/strong> . Pode ser que poderes pares sejam de um lado e poderes \u00edmpares de outro.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\"><strong>Varia\u00e7\u00e3o de sinais.<\/strong> Por exemplo, pode acontecer que os elementos de pot\u00eancias pares sejam positivos e os elementos de pot\u00eancias \u00edmpares sejam negativos, ou vice-versa.<\/span><\/li>\n<li style=\"margin-bottom:16px;\"> <span style=\"color:#000000;font-weight: normal;\"><strong>Repeti\u00e7\u00e3o:<\/strong> se a mesma matriz se repete a cada determinado n\u00famero de pot\u00eancias ou n\u00e3o.<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">Devemos tamb\u00e9m verificar se existe uma <strong>rela\u00e7\u00e3o<\/strong> entre o expoente e os elementos da matriz.<\/span> <\/li>\n<\/ul>\n<\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Exemplo de c\u00e1lculo da pot\u00eancia n de uma matriz:<\/h3>\n<ul>\n<li> Ser\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> a seguinte matriz, calcule<\/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> E<\/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> Vamos primeiro calcular v\u00e1rias pot\u00eancias da matriz<\/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> , para tentar adivinhar o padr\u00e3o seguido pelas pot\u00eancias. Ent\u00e3o calculamos<\/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> E <\/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=\"exerc\u00edcio resolvido passo a passo das pot\u00eancias de matrizes 2x2\" width=\"409\" height=\"361\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Ao calcular at\u00e9<\/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> , vemos que as pot\u00eancias da matriz<\/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> Elas seguem um padr\u00e3o: para cada aumento de pot\u00eancia, o resultado \u00e9 multiplicado por 2. Portanto, <strong>todas as matrizes s\u00e3o pot\u00eancias de 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> Podemos, portanto, derivar a f\u00f3rmula para a <strong>en\u00e9sima pot\u00eancia<\/strong> da matriz <\/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=\"en\u00e9sima pot\u00eancia de uma matriz 2x2\" width=\"201\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> E a partir desta f\u00f3rmula podemos calcular <\/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=\"exerc\u00edcio resolvido passo a passo pot\u00eancia de uma matriz 2x2\" width=\"187\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"> Problemas de pot\u00eancia de matriz resolvidos<\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Considere a seguinte matriz de dimens\u00e3o 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> Calcular: <\/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>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para calcular a pot\u00eancia de uma matriz, voc\u00ea deve multiplicar a matriz uma por uma. Portanto, primeiro calculamos <\/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\"> Agora calculamos <\/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\"> E finalmente calculamos <\/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\">Exerc\u00edcio 2<\/h3>\n<p> Considere a seguinte matriz de ordem 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> Calcular: <\/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>veja solu\u00e7\u00e3o<\/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> \u00e9 uma pot\u00eancia muito grande para ser calculada manualmente, ent\u00e3o as pot\u00eancias da matriz devem seguir um padr\u00e3o. Ent\u00e3o vamos calcular<\/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> para tentar entender a sequ\u00eancia que eles seguem: <\/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\"> Desta forma podemos ver o padr\u00e3o que as pot\u00eancias seguem: a cada pot\u00eancia, todos os n\u00fameros permanecem iguais, exceto o elemento da segunda coluna da segunda linha, que \u00e9 multiplicado por 3. Portanto, <strong>todos os n\u00fameros permanecem sempre iguais. e o \u00faltimo elemento \u00e9 uma pot\u00eancia de 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\"> Portanto, a f\u00f3rmula para <strong>a en\u00e9sima pot\u00eancia da matriz<\/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> Leste:<\/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\"> E a partir desta f\u00f3rmula podemos calcular <\/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\">Exerc\u00edcio 3<\/h3>\n<p> Considere a seguinte matriz 3\u00d73:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-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> Calcular: <\/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>veja solu\u00e7\u00e3o<\/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> \u00e9 uma pot\u00eancia muito grande para ser calculada manualmente, ent\u00e3o as pot\u00eancias da matriz devem seguir um padr\u00e3o. Ent\u00e3o vamos calcular<\/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> para tentar entender a sequ\u00eancia que eles seguem: <\/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\"> Desta forma podemos ver o padr\u00e3o que as pot\u00eancias seguem: a cada pot\u00eancia, todos os n\u00fameros permanecem iguais, exceto as fra\u00e7\u00f5es, que <strong>aumentam uma unidade no numerador:<\/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\"> Portanto, a f\u00f3rmula para <strong>a pot\u00eancia da <strong>en\u00e9sima<\/strong> matriz<\/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> Leste:<\/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\"> E a partir desta f\u00f3rmula podemos calcular <\/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\"> Exerc\u00edcio 4<\/h3>\n<p> Considere a seguinte matriz de tamanho 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> Calcular: <\/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>veja solu\u00e7\u00e3o<\/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> \u00e9 uma pot\u00eancia muito grande para ser calculada manualmente, ent\u00e3o as pot\u00eancias da matriz devem seguir um padr\u00e3o. Neste caso \u00e9 necess\u00e1rio calcular<\/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> para saber a sequ\u00eancia que seguem: <\/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\"> Com esses c\u00e1lculos podemos ver que a cada 4 pot\u00eancias obtemos a matriz identidade. Isso quer dizer que nos dar\u00e1 como resultado a matriz identidade dos poderes<\/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 Ent\u00e3o, para calcular<\/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> devemos decompor 201 em m\u00faltiplos de 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=\"exerc\u00edcio resolvido passo a passo das pot\u00eancias de matrizes 2x2 e pot\u00eancia 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> ,Ainda,<\/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> ser\u00e3o 50 vezes<\/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> e uma vez<\/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\"> E como sabemos disso<\/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> \u00e9 a matriz identidade <\/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\"> Al\u00e9m disso, a matriz identidade elevada a qualquer n\u00famero fornece a matriz identidade. Ainda:<\/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\"> E, finalmente, qualquer matriz multiplicada pela matriz identidade d\u00e1 a mesma matriz. ENT\u00c3O:<\/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\"> Para que<\/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> \u00e9 igual a <\/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\">Exerc\u00edcio 5<\/h3>\n<p> Considere a seguinte matriz de ordem 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> Calcular: <\/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>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Obviamente, calcule a pot\u00eancia da matriz<\/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> Este \u00e9 um c\u00e1lculo muito grande para ser feito manualmente, ent\u00e3o as pot\u00eancias da matriz devem seguir um padr\u00e3o. Neste caso \u00e9 necess\u00e1rio calcular<\/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> para saber a sequ\u00eancia que seguem: <\/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\"> Com esses c\u00e1lculos podemos ver que a cada 3 pot\u00eancias obtemos a matriz identidade. Isso quer dizer que nos dar\u00e1 como resultado a matriz identidade dos poderes<\/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> ,\u2026 Ent\u00e3o isso para calcular<\/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> Devemos decompor 62 em m\u00faltiplos de 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=\"exerc\u00edcio resolvido passo a passo de uma pot\u00eancia de uma matriz 3x3, en\u00e9sima pot\u00eancia\" 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> ,Ainda,<\/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> ser\u00e3o 20 vezes<\/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> e uma vez<\/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\"> E como sabemos disso<\/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> \u00e9 a matriz identidade <\/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\"> Al\u00e9m disso, a matriz identidade elevada a qualquer n\u00famero fornece a matriz identidade. Ainda:<\/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\"> Finalmente, qualquer matriz multiplicada pela matriz identidade d\u00e1 a mesma matriz. Ainda:<\/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\"> Para que<\/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> ser\u00e1 igual a<\/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> , para o qual calculamos o resultado anteriormente:<\/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> Se esses exerc\u00edcios sobre pot\u00eancias de matrizes quadradas foram \u00fateis para voc\u00ea, voc\u00ea tamb\u00e9m pode encontrar exerc\u00edcios passo a passo resolvidos sobre adi\u00e7\u00e3o e <a href=\"https:\/\/mathority.org\/pt\/adicao-subtracao-de-matrizes-2x2-3x3-exemplos-exercicios-resolvidos\/\">subtra\u00e7\u00e3o de matrizes<\/a> , uma das opera\u00e7\u00f5es com matrizes mais utilizadas.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina veremos como fazer pot\u00eancias de matrizes. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos passo a passo de pot\u00eancias de matrizes que o ajudar\u00e3o a entend\u00ea-lo perfeitamente. Voc\u00ea tamb\u00e9m aprender\u00e1 o que \u00e9 a en\u00e9sima pot\u00eancia de uma matriz e como encontr\u00e1-la. Como \u00e9 calculada a pot\u00eancia de uma matriz? Para calcular a pot\u00eancia &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/potencias-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Poderes da matriz<\/span> Leia mais &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":[12],"tags":[],"class_list":["post-282","post","type-post","status-publish","format-standard","hentry","category-determinante-de-uma-matriz"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Poderes da matriz -<\/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\/pt\/potencias-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Poderes da matriz -\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina veremos como fazer pot\u00eancias de matrizes. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos passo a passo de pot\u00eancias de matrizes que o ajudar\u00e3o a entend\u00ea-lo perfeitamente. Voc\u00ea tamb\u00e9m aprender\u00e1 o que \u00e9 a en\u00e9sima pot\u00eancia de uma matriz e como encontr\u00e1-la. Como \u00e9 calculada a pot\u00eancia de uma matriz? 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