{"id":281,"date":"2023-07-06T21:48:53","date_gmt":"2023-07-06T21:48:53","guid":{"rendered":"https:\/\/mathority.org\/pt\/multiplicacao-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos-passo-a-passo\/"},"modified":"2023-07-06T21:48:53","modified_gmt":"2023-07-06T21:48:53","slug":"multiplicacao-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos-passo-a-passo","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/multiplicacao-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos-passo-a-passo\/","title":{"rendered":"Multiplica\u00e7\u00e3o da matriz"},"content":{"rendered":"<p>Nesta p\u00e1gina veremos como <strong>multiplicar matrizes<\/strong> de dimens\u00f5es 2\u00d72, 3\u00d73, 4\u00d74, etc. Explicamos passo a passo o procedimento de multiplica\u00e7\u00e3o de matrizes atrav\u00e9s de um exemplo, a seguir voc\u00ea encontrar\u00e1 exerc\u00edcios resolvidos para que tamb\u00e9m possa praticar. Finalmente, voc\u00ea aprender\u00e1 quando duas matrizes n\u00e3o podem ser multiplicadas e todas as propriedades desta opera\u00e7\u00e3o matricial.<\/p>\n<h2 class=\"wp-block-heading\"> Como multiplicar duas matrizes?<\/h2>\n<p> Vejamos o procedimento para realizar a multiplica\u00e7\u00e3o de duas matrizes com um exemplo: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-multiplication-matricielle-22152.webp\" alt=\"exemplo de como multiplicar duas matrizes de dimens\u00e3o 2x2, opera\u00e7\u00f5es com matrizes\" width=\"228\" height=\"60\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Para calcular uma <strong>multiplica\u00e7\u00e3o de matrizes,<\/strong> as <strong>linhas<\/strong> da matriz esquerda devem ser multiplicadas pelas <strong>colunas<\/strong> da matriz direita.<\/p>\n<p> Ent\u00e3o, primeiro precisamos multiplicar <strong>a primeira linha pela primeira coluna.<\/strong> Para fazer isso, multiplicamos cada elemento da primeira linha por cada elemento da primeira coluna, um por um, e somamos os resultados. Ent\u00e3o tudo isso ser\u00e1 o primeiro elemento da primeira linha do array resultante. Veja o procedimento: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/comment-multiplier-des-matrices-22152.webp\" alt=\"como resolver multiplica\u00e7\u00e3o de matrizes 2x2, opera\u00e7\u00f5es com matrizes\" width=\"504\" height=\"87\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> 1 <strong>\u22c5<\/strong> 3 + 2 <strong>\u22c5<\/strong> 4 = 3 + 8 = 11. Ent\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p class=\"has-text-align-justify\"> Agora precisamos multiplicar <strong>a primeira linha pela segunda coluna<\/strong> . Repetimos, portanto, o procedimento: multiplicamos cada elemento da primeira linha um por um por cada elemento da segunda coluna e somamos os resultados. E tudo isso ser\u00e1 o segundo elemento da primeira linha do array resultante:<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p>1 <strong>\u22c5<\/strong> 5 + 2 <strong>\u22c5<\/strong> 1 = 5 + 2 = 7. Ent\u00e3o: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Depois de preencher a primeira linha da matriz resultante, passamos para a segunda linha. Multiplicamos, portanto <strong>, a segunda linha pela primeira coluna<\/strong> , repetindo o procedimento: multiplicamos um por um cada elemento da segunda linha por cada elemento da primeira coluna e somamos os resultados:<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p>-3 <strong>\u22c5<\/strong> 3 + 0 <strong>\u22c5<\/strong> 4 = -9 + 0 = -9. Ainda: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p class=\"has-text-align-justify\"> Finalmente, multiplicamos <strong>a segunda linha pela segunda coluna<\/strong> . Sempre com o mesmo procedimento: multiplicamos cada elemento da segunda linha um por um por cada elemento da segunda coluna e somamos os resultados:<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p>-3 <strong>\u22c5<\/strong> 5 + 0 <strong>\u22c5<\/strong> 1 = -15 + 0 = -15. Ainda:<\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p>E aqui termina a multiplica\u00e7\u00e3o das duas matrizes. Como voc\u00ea viu, \u00e9 necess\u00e1rio multiplicar as linhas pelas colunas, repetindo sempre o mesmo procedimento: multiplicar cada elemento da linha por cada elemento da coluna um por um, e somar os resultados.<\/p>\n<h2 class=\"wp-block-heading\"> Exerc\u00edcios resolvidos de multiplica\u00e7\u00e3o de matrizes<\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Resolva o seguinte produto matricial: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-produit-de-matrices-22.webp\" alt=\"exerc\u00edcio resolvido produto passo a passo de matrizes 2x2, opera\u00e7\u00f5es com matrizes\" width=\"172\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 um produto de matrizes 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-747926b92c1d388c1150613b0f471d7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"142\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para resolver um produto matricial, voc\u00ea deve multiplicar as linhas da matriz esquerda pelas colunas da matriz direita.<\/p>\n<p class=\"has-text-align-left has-text-align-justify\"> Ent\u00e3o primeiro multiplicamos <strong>a primeira linha pela primeira coluna.<\/strong> Para fazer isso, multiplicamos cada elemento da primeira linha por cada elemento da primeira coluna, um por um, e somamos os resultados. E tudo isso ser\u00e1 o primeiro elemento da primeira linha do array resultante:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eff23eaf91738d6ffb383949e4b70856_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix}  = \\begin{pmatrix} 1\\cdot 3 +2 \\cdot 1 &amp; \\\\[1.1ex] &amp; \\end{pmatrix} = \\begin{pmatrix} 5 &amp; \\\\[1.1ex] &amp; \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"370\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos multiplicar <strong>a primeira linha pela segunda coluna,<\/strong> para obter o segundo elemento da primeira linha da matriz resultante:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-558838bcc38efc1aeeaf298d3e7151dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix}  = \\begin{pmatrix} -1 &amp; 1\\cdot (-2) +2 \\cdot 5 \\\\[1.1ex] &amp; \\end{pmatrix} = \\begin{pmatrix}5 &amp; 8 \\\\[1.1ex] &amp; \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"429\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Vamos para a segunda linha, ent\u00e3o multiplicamos <strong>a segunda linha pela primeira coluna:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-daab54a49cc53c320bb2965f691fd7ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix} = \\begin{pmatrix} -1 &amp; 8 \\\\[1.1ex] 3\\cdot 3 +4 \\cdot 1 &amp; \\end{pmatrix}= \\begin{pmatrix}5 &amp; 8 \\\\[1.1ex] 13 &amp; \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"396\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por fim, multiplicamos <strong>a segunda linha pela segunda coluna<\/strong> , para calcular o \u00faltimo elemento da tabela:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a85e0d62a0db18c7712fd1b354f92bd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 2 \\\\[1.1ex] 3 &amp; 4  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 1 &amp; 5  \\end{pmatrix}= \\begin{pmatrix} -1 &amp; 8 \\\\[1.1ex]1 &amp; 3\\cdot (-2) +4 \\cdot 5 \\end{pmatrix}=\\begin{pmatrix} 5 &amp; 8 \\\\[1.1ex] 13 &amp; 14 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"447\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, o resultado da multiplica\u00e7\u00e3o de matrizes \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-76f1283db0175bc1a95b0a10c8961761_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} \\bm{5} &amp; \\bm{8} \\\\[1.1ex]\\bm{13} &amp; \\bm{14} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"72\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Encontre o resultado da seguinte multiplica\u00e7\u00e3o de matriz quadrada 2&#215;2: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-multiplication-matricielle-22.webp\" alt=\"Exerc\u00edcio resolvido passo a passo em multiplica\u00e7\u00e3o de matrizes 2x2, opera\u00e7\u00f5es matriciais\" width=\"230\" height=\"70\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 um produto de matrizes de dimens\u00e3o 2\u00d72.<\/p>\n<p class=\"has-text-align-left\"> Para resolver a multiplica\u00e7\u00e3o, voc\u00ea deve multiplicar as linhas da matriz esquerda pelas colunas da matriz direita: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fc7217dab49f67df2a9d2abc561baf9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{pmatrix} 4 &amp; -1  \\\\[1.1ex] -2 &amp; 3  \\end{pmatrix} \\cdot \\begin{pmatrix} -2 &amp; 5 \\\\[1.1ex] 6 &amp; -3  \\end{pmatrix}  &amp; = \\begin{pmatrix} 4\\cdot (-2)+(-1) \\cdot 6 &amp;  4\\cdot 5+(-1) \\cdot (-3)  \\\\[1.1ex](-2)\\cdot (-2)+3 \\cdot 6 &amp; (-2)\\cdot 5+3 \\cdot (-3)\\end{pmatrix} \\\\[2ex] &amp; =\\begin{pmatrix} \\bm{-14} &amp; \\bm{23} \\\\[1.1ex]\\bm{22} &amp; \\bm{-19} \\end{pmatrix} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"528\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 3<\/h3>\n<p> Calcule a seguinte multiplica\u00e7\u00e3o de matrizes 3&#215;3: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-de-multiplication-matricielle-33.webp\" alt=\"exerc\u00edcio resolvido multiplica\u00e7\u00e3o passo a passo de matrizes 3x3, opera\u00e7\u00f5es matriciais\" width=\"277\" height=\"109\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para realizar a multiplica\u00e7\u00e3o de matrizes 3\u00d73, voc\u00ea deve multiplicar as linhas da matriz esquerda pelas colunas da matriz direita: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef6ee7bb6e4ac095a9fd51a545b163b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{l} \\begin{pmatrix} 1 &amp; 2 &amp; 0 \\\\[1.1ex] 3 &amp; 2 &amp; -1 \\\\[1.1ex] 5 &amp; 1 &amp; -2  \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 &amp; 0 \\\\[1.1ex] 1 &amp; 0 &amp; -2 \\\\[1.1ex] -1 &amp; 2 &amp; 1 \\end{pmatrix} = \\\\[7.5ex] =\\begin{pmatrix} 1 \\cdot 3+2 \\cdot 1+ 0 \\cdot (-1) &amp; 1 \\cdot 4+2 \\cdot 0+ 0 \\cdot 2 &amp; 1 \\cdot 0+2 \\cdot (-2)+ 0 \\cdot 1 \\\\[1.1ex] 3 \\cdot 3+2 \\cdot 1+ (-1) \\cdot (-1) &amp; 3 \\cdot 4+2 \\cdot 0+ (-1) \\cdot 2 &amp; 3 \\cdot 0+2 \\cdot (-2)+ (-1) \\cdot 1 \\\\[1.1ex] 5 \\cdot 3+1 \\cdot 1+ (-2) \\cdot (-1) &amp; 5 \\cdot 4+1 \\cdot 0+ (-2) \\cdot 2 &amp; 5 \\cdot 0+1 \\cdot (-2)+ (-2) \\cdot 1 \\end{pmatrix} = \\\\[7.5ex]  =\\begin{pmatrix} \\bm{5} &amp; \\bm{4} &amp; \\bm{-4} \\\\[1.1ex] \\bm{12} &amp; \\bm{10} &amp; \\bm{-5} \\\\[1.1ex] \\bm{18} &amp; \\bm{16} &amp; \\bm{-4} \\end{pmatrix}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"306\" width=\"643\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 4<\/h3>\n<p> dada a 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> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-27365f9993caf4fcdb747352e4ae539d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} 3 &amp; 1 &amp; -2 \\\\[1.1ex] 4 &amp; 2 &amp; -1   \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"134\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> 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-307d37497055a6891b797bdb89b456e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2A\\cdot A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Vamos primeiro calcular a matriz transposta de<\/p>\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 fazer a multiplica\u00e7\u00e3o. E para fazer a matriz de transposi\u00e7\u00e3o, precisamos transformar as linhas em colunas. Ou seja, a primeira linha da matriz torna-se a primeira coluna da matriz e a segunda linha da matriz torna-se a segunda coluna da 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-ac4785c47f2e48e15b3d98ba426848b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^t= \\begin{pmatrix} 3 &amp; 4 \\\\[1.1ex] 1 &amp; 2  \\\\[1.1ex] -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"131\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A opera\u00e7\u00e3o matricial, portanto, permanece:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9513fa8cc6996e18e3cf287f0210817a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2A\\cdot A^t = 2 \\begin{pmatrix} 3 &amp; 1 &amp; -2 \\\\[1.1ex] 4 &amp; 2 &amp; -1   \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 \\\\[1.1ex] 1 &amp; 2  \\\\[1.1ex] -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"291\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora podemos fazer os c\u00e1lculos. 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-d4e94385e2fa1b091190a9ce266a8c43_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\"><\/p>\n<p> (embora tamb\u00e9m possamos primeiro calcular<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae92cabff7a388b31fe67b559dfead7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A \\cdot A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"44\" style=\"vertical-align: 0px;\"><\/p>\n<p> ): <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae5e95f09aedac8f0861bf13fb9c78a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 2 \\cdot 3 &amp; 2 \\cdot 1 &amp; 2 \\cdot (-2) \\\\[1.1ex] 2 \\cdot 4 &amp; 2 \\cdot 2 &amp; 2 \\cdot (-1) \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 \\\\[1.1ex] 1 &amp; 2  \\\\[1.1ex] -2 &amp; -1 \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"299\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-24c003b8da1081d6ca494adc3356b06b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  =\\begin{pmatrix} 6 &amp; 2 &amp; -4 \\\\[1.1ex] 8 &amp; 4 &amp; -2 \\end{pmatrix} \\cdot \\begin{pmatrix} 3 &amp; 4 \\\\[1.1ex] 1 &amp; 2  \\\\[1.1ex] -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"220\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, finalmente, resolvemos o produto das matrizes: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0eb8f1817f0163a82ae39cc6c81d478e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 6 \\cdot 3 +2 \\cdot 1 + (-4) \\cdot (-2) &amp; 6 \\cdot 4 +2 \\cdot 2 + (-4) \\cdot (-1) \\\\[1.1ex] 8 \\cdot 3 +4 \\cdot 1 + (-2) \\cdot (-2) &amp; 8 \\cdot 4 +4 \\cdot 2 + (-2) \\cdot (-1) \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"438\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33533be747b72497915048e486d16541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = \\begin{pmatrix} \\bm{28} &amp; \\bm{32} \\\\[1.1ex]\\bm{32} &amp; \\bm{42} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"94\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 5<\/h3>\n<p> Considere as seguintes matrizes:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6e26aec2eee6bcae0e344682d20038f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix} \\qquad B=\\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"275\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> 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-78d69cf0ef5ec44cd0aacf00f4f2d613_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot B - B \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"102\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 uma opera\u00e7\u00e3o que combina subtra\u00e7\u00e3o com multiplica\u00e7\u00f5es de matrizes 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-43f79f2d970bb02caaeddec34d5ad2a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot B - B \\cdot A= \\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix}\\cdot \\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix} - \\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}  \\cdot \\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"500\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Primeiro calculamos a multiplica\u00e7\u00e3o \u00e0 esquerda: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05ff586671fb0af274884169c54e5817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2\\cdot (-1) + 4 \\cdot 3 &amp; 2\\cdot (-2) + 4 \\cdot (-3) \\\\[1.1ex] (-3)\\cdot (-1) + 5 \\cdot 3 &amp; (-3)\\cdot (-2) + 5 \\cdot (-3)  \\end{pmatrix} - \\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}  \\cdot \\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"550\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-43c234a2d7aa4f9dcaf3140f617480f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle= \\begin{pmatrix} 10 &amp; -16  \\\\[1.1ex] 18 &amp; -9 \\end{pmatrix} - \\begin{pmatrix} -1 &amp; -2  \\\\[1.1ex] 3 &amp; -3 \\end{pmatrix}  \\cdot \\begin{pmatrix} 2 &amp; 4  \\\\[1.1ex] -3 &amp; 5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"308\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora resolvemos a multiplica\u00e7\u00e3o \u00e0 direita: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-552309dd1be2f69bb72633539809283b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 10 &amp; -16  \\\\[1.1ex] 18 &amp; -9 \\end{pmatrix} - \\begin{pmatrix} -1 \\cdot 2 +(-2) \\cdot (-3) &amp;  -1 \\cdot 4 +(-2) \\cdot 5  \\\\[1.1ex]3 \\cdot 2 +(-3) \\cdot (-3) &amp;  3 \\cdot 4 +(-3) \\cdot 5  \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"449\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eeac84965cc522402e869234a841ba67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle =\\begin{pmatrix} 10 &amp; -16  \\\\[1.1ex] 18 &amp; -9 \\end{pmatrix} - \\begin{pmatrix} 4 &amp;-14  \\\\[1.1ex]15 &amp; -3  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"223\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente subtra\u00edmos as matrizes: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-faefbc14fc49439616b3d131243eba79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 10-4 &amp; -16 -(-14) \\\\[1.1ex] 18-15 &amp; -9-(-3) \\end{pmatrix} =\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"214\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-50bac6ac99e1cf6e4b77a1a8718f9fe4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle =\\begin{pmatrix} \\bm{6} &amp; \\bm{-2} \\\\[1.1ex] \\bm{3} &amp; \\bm{-6} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"90\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\">Quando voc\u00ea n\u00e3o pode multiplicar duas matrizes?<\/h2>\n<p> <strong>Nem todas as matrizes podem ser multiplicadas.<\/strong> Para multiplicar duas matrizes, o n\u00famero de colunas da primeira matriz deve corresponder ao n\u00famero de linhas da segunda matriz.<\/p>\n<p> Por exemplo, a seguinte multiplica\u00e7\u00e3o n\u00e3o pode ser realizada porque a primeira matriz possui 3 colunas e a segunda matriz possui 2 linhas:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8314f9238afb3676bee5c9000c02752_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{pmatrix} 1 &amp; 3 &amp; -2 \\\\[1.1ex] 4 &amp; 0 &amp; 5 \\end{pmatrix} \\cdot  \\begin{pmatrix} 2 &amp; 1  \\\\[1.1ex] 3 &amp; -1  \\end{pmatrix}  \\ \\longleftarrow \\ \\color{red} \\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"274\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Mas se invertermos a ordem, eles podem ser multiplicados. Porque a primeira matriz possui duas colunas e a segunda matriz possui duas linhas:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-37d01cc99b578d3756312c3e6ff12cae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{pmatrix} 2 &amp; 1  \\\\[1.1ex] 3 &amp; -1  \\end{pmatrix} \\cdot \\begin{pmatrix} 1 &amp; 3 &amp; -2 \\\\[1.1ex] 4 &amp; 0 &amp; 5  \\end{pmatrix}  &amp; = \\begin{pmatrix} 2\\cdot 1 + 1 \\cdot 4 &amp; 2\\cdot 3 + 1 \\cdot 0 &amp; 2\\cdot (-2) + 1 \\cdot 5  \\\\[1.1ex] 3\\cdot 1 + (-1) \\cdot 4 &amp; 3\\cdot 3 + (-1) \\cdot 0 &amp; 3\\cdot (-2) + (-1) \\cdot 5   \\end{pmatrix} \\\\[2ex] &amp; = \\begin{pmatrix} \\bm{6} &amp; \\bm{6} &amp; \\bm{1}  \\\\[1.1ex]\\bm{-1} &amp; \\bm{9} &amp; \\bm{-11}   \\end{pmatrix}   \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"624\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Propriedades de multiplica\u00e7\u00e3o de matrizes<\/h2>\n<p> Este tipo de opera\u00e7\u00e3o matricial possui as seguintes caracter\u00edsticas:<\/p>\n<ul>\n<li> A multiplica\u00e7\u00e3o de matrizes \u00e9 <strong><span style=\"color:#1976d2;\">associativa:<\/span><\/strong><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-38541ff37ecadb79ac36ffb1e19cc187_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left( A \\cdot B \\right) \\cdot C = A \\cdot \\left( B \\cdot C \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> A multiplica\u00e7\u00e3o de matrizes tamb\u00e9m tem a <strong><span style=\"color:#1976d2;\">propriedade distributiva:<\/span><\/strong> <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f8ca2784a9dd93cf71cd34d4d0303eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot \\left(B+C\\right) = A\\cdot B + A \\cdot C\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"216\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<ul>\n<li> O produto de matrizes <strong><span style=\"color:#1976d2;\">n\u00e3o \u00e9 comutativo:<\/span><\/strong><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-67f2cce38b1aab5659a5f888daf1ff84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A \\cdot B \\neq B \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"104\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Por exemplo, a seguinte multiplica\u00e7\u00e3o de matrizes d\u00e1 um resultado:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e780b321b160ad4a612e608199a374b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{pmatrix} 1 &amp; -1  \\\\[1.1ex] 2 &amp; 3  \\end{pmatrix} \\cdot \\begin{pmatrix} -2 &amp; 5  \\\\[1.1ex] 0 &amp; 1   \\end{pmatrix}  &amp; = \\begin{pmatrix} 1\\cdot (-2) + (-1) \\cdot 0 &amp; 1\\cdot 5 + (-1) \\cdot 1   \\\\[1.1ex] 2\\cdot (-2) + 3 \\cdot 0 &amp;  2\\cdot 5 + 3 \\cdot 1    \\end{pmatrix} \\\\[2ex] &amp; = \\begin{pmatrix} \\bm{-2} &amp; \\bm{4} \\\\[1.1ex] \\bm{-4} &amp;  \\bm{13} \\end{pmatrix}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"472\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Mas o resultado do produto \u00e9 diferente se invertermos a ordem de multiplica\u00e7\u00e3o das matrizes:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-177d78a209e5d9e18828617e4913176d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned}\\begin{pmatrix} -2 &amp; 5  \\\\[1.1ex] 0 &amp; 1   \\end{pmatrix} \\cdot  \\begin{pmatrix} 1 &amp; -1  \\\\[1.1ex] 2 &amp; 3  \\end{pmatrix} &amp; = \\begin{pmatrix} -2 \\cdot 1 + 5\\cdot 2 &amp;  -2 \\cdot (-1) + 5\\cdot 3  \\\\[1.1ex] 0 \\cdot 1 + 1\\cdot 2 &amp;  0 \\cdot (-1) + 1\\cdot 3   \\end{pmatrix} \\\\[2ex] &amp; = \\begin{pmatrix} \\bm{8} &amp;  \\bm{17}  \\\\[1.1ex] \\bm{2} &amp;  \\bm{3} \\end{pmatrix}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"445\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Al\u00e9m disso, qualquer matriz multiplicada pela matriz identidade resulta na mesma matriz. Isso \u00e9 chamado <strong><span style=\"color:#1976d2;\">de propriedade de identidade multiplicativa:<\/span><\/strong><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7ab05972282922f1e10f75a50e636887_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A \\cdot I=A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"72\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c32986a7c34108a47500a4f0ec2967b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle I \\cdot A=A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"72\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> 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-9c1e72173419eb76554256cf6ccd0d2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 2 &amp; 7  \\\\[1.1ex] -6 &amp; 5  \\end{pmatrix} \\cdot \\begin{pmatrix} 1 &amp; 0  \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix} = \\begin{pmatrix} \\bm{2} &amp; \\bm{7}  \\\\[1.1ex] \\bm{-6} &amp; \\bm{5}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"242\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Finalmente, como voc\u00ea j\u00e1 deve imaginar, qualquer matriz multiplicada pela matriz zero \u00e9 igual \u00e0 matriz zero. Isso \u00e9 chamado <strong><span style=\"color:#1976d2;\">de propriedade multiplicativa de zero:<\/span><\/strong><\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cf700c38f25e0c3bdf1c46851341a815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A \\cdot 0=0\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac3340bc96ba3df60f6ddeb6bbd3b4b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 0\\cdot A=0\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> 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-3152d82054a80d61d548e969290aea4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 6 &amp; -4  \\\\[1.1ex] 3 &amp; 8  \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; 0  \\\\[1.1ex] 0 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} \\bm{0} &amp; \\bm{0}  \\\\[1.1ex] \\bm{0} &amp; \\bm{0}\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"228\" style=\"vertical-align: 0px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina veremos como multiplicar matrizes de dimens\u00f5es 2\u00d72, 3\u00d73, 4\u00d74, etc. Explicamos passo a passo o procedimento de multiplica\u00e7\u00e3o de matrizes atrav\u00e9s de um exemplo, a seguir voc\u00ea encontrar\u00e1 exerc\u00edcios resolvidos para que tamb\u00e9m possa praticar. Finalmente, voc\u00ea aprender\u00e1 quando duas matrizes n\u00e3o podem ser multiplicadas e todas as propriedades desta opera\u00e7\u00e3o matricial. Como &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/multiplicacao-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos-passo-a-passo\/\"> <span class=\"screen-reader-text\">Multiplica\u00e7\u00e3o 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":[25],"tags":[],"class_list":["post-281","post","type-post","status-publish","format-standard","hentry","category-pinturas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Multiplica\u00e7\u00e3o de matrizes - Matoridade<\/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\/multiplicacao-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos-passo-a-passo\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Multiplica\u00e7\u00e3o de matrizes - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina veremos como multiplicar matrizes de dimens\u00f5es 2\u00d72, 3\u00d73, 4\u00d74, etc. Explicamos passo a passo o procedimento de multiplica\u00e7\u00e3o de matrizes atrav\u00e9s de um exemplo, a seguir voc\u00ea encontrar\u00e1 exerc\u00edcios resolvidos para que tamb\u00e9m possa praticar. 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