{"id":279,"date":"2023-07-06T22:09:09","date_gmt":"2023-07-06T22:09:09","guid":{"rendered":"https:\/\/mathority.org\/pt\/multiplicacao-de-um-numero-por-uma-matriz-2x2-e-3x3-exemplos-e-exercicios-resolvidos\/"},"modified":"2023-07-06T22:09:09","modified_gmt":"2023-07-06T22:09:09","slug":"multiplicacao-de-um-numero-por-uma-matriz-2x2-e-3x3-exemplos-e-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/multiplicacao-de-um-numero-por-uma-matriz-2x2-e-3x3-exemplos-e-exercicios-resolvidos\/","title":{"rendered":"Multiplicando um n\u00famero por uma matriz"},"content":{"rendered":"<p>Nesta p\u00e1gina veremos como <strong>multiplicar um n\u00famero por uma matriz.<\/strong> Voc\u00ea tamb\u00e9m tem exemplos que o ajudar\u00e3o a entend\u00ea-lo perfeitamente e exerc\u00edcios resolvidos para que voc\u00ea possa praticar. Voc\u00ea tamb\u00e9m encontrar\u00e1 todas as propriedades do produto de um escalar e de uma matriz.<\/p>\n<h2 class=\"wp-block-heading\"> Como multiplicar um n\u00famero por uma matriz?<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Para <strong>multiplicar um n\u00famero por uma matriz<\/strong> , multiplique cada elemento da matriz pelo n\u00famero.<\/p>\n<h2 class=\"wp-block-heading\"> Exemplo: <\/h2>\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-dun-nombre-par-une-matrice.webp\" alt=\"exemplo de multiplica\u00e7\u00e3o ou produto de um n\u00famero por uma matriz\" width=\"514\" height=\"122\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"> Problemas resolvidos de multiplica\u00e7\u00e3o de um n\u00famero por uma matriz<\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1: <\/h3>\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-scalaire-entre-un-nombre-et-une-matrice-22.webp\" alt=\"Exerc\u00edcio resolvido do produto de um n\u00famero por uma matriz 2x2, opera\u00e7\u00f5es com matrizes\" width=\"107\" 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 uma multiplica\u00e7\u00e3o de um escalar por uma matriz quadrada 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-590b79c0fea524b963397181b6f2bea8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 3 \\begin{pmatrix} 1 &amp; 3 \\\\[1.1ex] 2 &amp; -4  \\end{pmatrix} = \\begin{pmatrix} 3\\cdot 1 &amp; 3\\cdot 3 \\\\[1.1ex] 3\\cdot 2 &amp; 3\\cdot (-4)  \\end{pmatrix} = \\begin{pmatrix} \\bm{3} &amp; \\bm{9} \\\\[1.1ex] \\bm{6} &amp; \\bm{-12} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"348\" 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<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-dun-nombre-par-une-matrice-33.webp\" alt=\"exerc\u00edcio resolvido passo a passo de multiplica\u00e7\u00e3o de um n\u00famero por uma matriz 3x3, opera\u00e7\u00f5es com matrizes\" width=\"184\" height=\"106\" 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 o produto de um n\u00famero por uma matriz quadrada 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-5042f0f8cd9b7a4d0e28974f793b145b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -4 \\begin{pmatrix} 2 &amp; 1 &amp; 5 \\\\[1.1ex] -1 &amp; 0 &amp; 3 \\\\[1.1ex] 6 &amp; -2 &amp; -3  \\end{pmatrix} = \\begin{pmatrix} -4 \\cdot 2 &amp; -4 \\cdot 1 &amp; -4 \\cdot 5 \\\\[1.1ex] -4 \\cdot (-1) &amp; -4 \\cdot 0 &amp; -4 \\cdot 3 \\\\[1.1ex] -4 \\cdot 6 &amp; -4 \\cdot (-2) &amp; -4 \\cdot (-3)  \\end{pmatrix}= \\begin{pmatrix} \\bm{-8} &amp; \\bm{-4} &amp; \\bm{-20} \\\\[1.1ex] \\bm{4} &amp; \\bm{0} &amp; \\bm {-12}  \\\\[1.1ex] \\bm{-24} &amp; \\bm{8} &amp; \\bm {12} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"627\" 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<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\/operations-combinees-avec-des-matrices-22152.webp\" alt=\"Exerc\u00edcio resolvido de multiplica\u00e7\u00e3o de um n\u00famero por uma matriz 2x2, opera\u00e7\u00f5es combinadas com matrizes\" width=\"233\" 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 uma opera\u00e7\u00e3o que combina produtos de n\u00fameros por matrizes e somas de matrizes 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-56d2a40f021be13a5d92d0c10d353684_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2 \\begin{pmatrix} 5 &amp; 1 \\\\[1.1ex] -2 &amp; 3  \\end{pmatrix}+5\\begin{pmatrix} 5 &amp; 1 \\\\[1.1ex] -2 &amp; 3  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"193\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, primeiro precisamos resolver os produtos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-068901abef987767025bb01b24579226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 10 &amp; 2 \\\\[1.1ex] -4 &amp; 6  \\end{pmatrix}+\\begin{pmatrix} 25 &amp; 5 \\\\[1.1ex] -10 &amp; 15  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"183\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente adicionamos as matrizes resultantes: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d15ea16036f522af0f23fee0bb796757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} \\bm{35} &amp; \\bm{7} \\\\[1.1ex] \\bm{-14} &amp; \\bm{21}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"85\" 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 class=\"has-text-align-left\"> 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-5374cb55dbbfa80c91a478b4cbdb2ee1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; -3 &amp; 5 \\\\ 1 &amp; 4 &amp; 0 \\\\ -3 &amp; 2 &amp; -5 \\end{pmatrix}  \\qquad B=\\begin{pmatrix} 6 &amp; 0 &amp; 2 \\\\ -3 &amp; 4 &amp; 1 \\\\ 3 &amp; 2 &amp; 7 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"344\" style=\"vertical-align: -27px;\"><\/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-d5f0b93a77e7bb1b7b99d63546652e8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2A+5I-3B\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"119\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> \u00c9 uma opera\u00e7\u00e3o que combina multiplica\u00e7\u00f5es escalares com adi\u00e7\u00f5es e subtra\u00e7\u00f5es de matrizes de dimens\u00e3o 3\u00d73. Al\u00e9m disso, 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-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"I\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 a matriz identidade, que \u00e9 composta por 1 na diagonal principal e 0 nos demais elementos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dce934040dc05714321dbbeac4e20c73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2\\begin{pmatrix} 2 &amp; -3 &amp; 5 \\\\[1.1ex] 1 &amp; 4 &amp; 0 \\\\[1.1ex] -3 &amp; 2 &amp; -5 \\end{pmatrix}+5\\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\[1.1ex]  0 &amp; 1 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 1 \\end{pmatrix} -3 \\begin{pmatrix} 6 &amp; 0 &amp; 2 \\\\[1.1ex] -3 &amp; 4 &amp; 1 \\\\[1.1ex] 3 &amp; 2 &amp; 7 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"412\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, primeiro realizamos as multiplica\u00e7\u00f5es:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc26f29384abcfb6f08a36b601e4ff61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -4 &amp; 6 &amp; -10 \\\\[1.1ex] -2 &amp; -8 &amp; 0 \\\\[1.1ex] 6 &amp; -4 &amp; 10 \\end{pmatrix}+\\begin{pmatrix} 5 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 5 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 5 \\end{pmatrix} - \\begin{pmatrix} 18 &amp; 0 &amp; 6 \\\\[1.1ex] -9 &amp; 12 &amp; 3 \\\\[1.1ex] 9 &amp; 6 &amp; 21 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"385\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Adicionamos as duas primeiras 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-897ec02d46bc09bdec58d9b3246c6f4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle   \\begin{pmatrix} 1 &amp; 6 &amp; -10 \\\\[1.1ex] -2 &amp; -3 &amp; 0 \\\\[1.1ex] 6 &amp; -4 &amp; 15 \\end{pmatrix}-\\begin{pmatrix} 18 &amp; 0 &amp; 6 \\\\[1.1ex] -9 &amp; 12 &amp; 3 \\\\[1.1ex] 9 &amp; 6 &amp; 21 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"274\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por fim, realizamos a subtra\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-9ddd808a46a137f4c7742545c4f76f46_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} \\bm{-17} &amp; \\bm{6} &amp; \\bm{-16} \\\\[1.1ex] \\bm{7} &amp; \\bm{-15} &amp; \\bm{-3} \\\\[1.1ex] \\bm{-3} &amp; \\bm{-10} &amp; \\bm{-6} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"148\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Se estes exerc\u00edcios sobre produtos escalares de matrizes foram \u00fateis para voc\u00ea, n\u00e3o hesite em praticar com os exerc\u00edcios resolvidos passo a passo sobre a <a href=\"https:\/\/mathority.org\/pt\/adicao-subtracao-de-matrizes-2x2-3x3-exemplos-exercicios-resolvidos\/\">adi\u00e7\u00e3o de matrizes<\/a> e o <a href=\"https:\/\/mathority.org\/pt\/multiplicacao-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos-passo-a-passo\/\">produto de matrizes<\/a> , os dois tipos de opera\u00e7\u00f5es matriciais que se repetem mais.<\/p>\n<h2 class=\"wp-block-heading\"> Propriedades do produto de um n\u00famero por uma matriz<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Como voc\u00ea bem sabe, existem muitos <a href=\"https:\/\/mathority.org\/pt\/tipos-de-matriz\/\">tipos de matrizes<\/a> : matrizes quadradas, matrizes triangulares, matriz identidade, etc. Mas, felizmente, todas as propriedades do produto de n\u00fameros por matrizes s\u00e3o v\u00e1lidas para todas as classes de matrizes.<\/p>\n<p> Aqui est\u00e3o as propriedades de multiplica\u00e7\u00e3o entre escalares e matrizes:<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Propriedade 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-bff3550cd8d240f651354e6646e6bf15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a \\cdot (b \\cdot A) = (a \\cdot b) \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"163\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Observe as duas opera\u00e7\u00f5es a seguir porque elas fornecem o mesmo resultado, n\u00e3o importa como multiplicamos 2 e 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-b4e9fd568edd5833238d8d21fdf4d1a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2 \\cdot \\left(3 \\cdot \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 2 &amp; -1 \\end{pmatrix} \\right) =2 \\cdot \\begin{pmatrix} 3 &amp; 0 \\\\[1.1ex] 6 &amp; -3 \\end{pmatrix} = \\begin{pmatrix} \\bm{6} &amp; \\bm{0} \\\\[1.1ex] \\bm{12} &amp; \\bm{-6} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"372\" 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-9f8ee596b3e2ca16ff1c507717982ee1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (2 \\cdot 3) \\cdot \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 2 &amp; -1 \\end{pmatrix}  =6 \\cdot \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 2 &amp; -1 \\end{pmatrix}   = \\begin{pmatrix} \\bm{6} &amp; \\bm{0} \\\\[1.1ex] \\bm{12} &amp; \\bm{-6}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"357\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Propriedade distributiva<\/span><\/strong> em rela\u00e7\u00e3o \u00e0 adi\u00e7\u00e3o de escalares:<\/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-5829e6e40633068cc4f35b43184a41e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b) \\cdot A = a \\cdot A+ b \\cdot A\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"192\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como voc\u00ea pode ver no exemplo abaixo, \u00e9 o mesmo se primeiro adicionarmos 1+2 e depois multiplicarmos por uma matriz, ou se multiplicarmos a matriz separadamente por 1 e por 2 e depois adicionarmos os resultados:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-025ac9b0851ed93fd0c3870328d6144b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (1 + 2) \\cdot  \\begin{pmatrix} 2 &amp; -1 \\\\[1.1ex] 3 &amp; 5 \\\\[1.1ex] -2 &amp; -4 \\end{pmatrix} =3 \\cdot  \\begin{pmatrix} 2 &amp; -1 \\\\[1.1ex] 3 &amp; 5 \\\\[1.1ex] -2 &amp; -4 \\end{pmatrix}=  \\begin{pmatrix} \\bm{6} &amp; \\bm{-3} \\\\[1.1ex] \\bm{9} &amp; \\bm{15} \\\\[1.1ex] \\bm{-6} &amp; \\bm{-12} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"416\" 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-2f54f4d5ae113e2462b752c150b3f43b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1  \\cdot  \\begin{pmatrix} 2 &amp; -1 \\\\[1.1ex] 3 &amp; 5 \\\\[1.1ex] -2 &amp; -4 \\end{pmatrix} + 2  \\cdot  \\begin{pmatrix} 2 &amp; -1 \\\\[1.1ex] 3 &amp; 5 \\\\[1.1ex] -2 &amp; -4\\end{pmatrix} = \\begin{pmatrix} 2 &amp; -1 \\\\[1.1ex] 3 &amp; 5\\\\[1.1ex] -2 &amp; -4 \\end{pmatrix} +  \\begin{pmatrix} 4 &amp; -2 \\\\[1.1ex] 6 &amp; 10 \\\\[1.1ex] -4 &amp; -8\\end{pmatrix}=  \\begin{pmatrix} \\bm{6} &amp; \\bm{-3} \\\\[1.1ex] \\bm{9} &amp; \\bm{15} \\\\[1.1ex] \\bm{-6} &amp; \\bm{-12}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"568\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Propriedade distributiva<\/span><\/strong> em rela\u00e7\u00e3o \u00e0 adi\u00e7\u00e3o de matrizes:<\/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-fac6ec8cbb2d4ead773b75d0180bca20_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a \\cdot \\left(A + B \\right) = a \\cdot A + a \\cdot B\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"202\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Em outras palavras, somar duas matrizes matem\u00e1ticas e depois multiplic\u00e1-las por um n\u00famero equivale a multiplicar separadamente as duas matrizes pelo mesmo n\u00famero e depois somar os resultados. No exemplo abaixo voc\u00ea pode verificar:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdb35d5c66ee525c3d52fe7576e75758_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 4 \\cdot  \\left( \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 6 &amp; -1 \\end{pmatrix}+\\begin{pmatrix} -1 &amp; 3 \\\\[1.1ex] 0 &amp; 4 \\end{pmatrix} \\right) =4 \\cdot   \\begin{pmatrix} 2 &amp; 1 \\\\[1.1ex] 6 &amp; 3 \\end{pmatrix}= \\begin{pmatrix} \\bm{8} &amp; \\bm{4} \\\\[1.1ex] \\bm{24} &amp; \\bm{12} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"430\" 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-0ef9d3f8f503371fa5f3d2478f728d88_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 4 \\cdot  \\begin{pmatrix} 3 &amp; -2 \\\\[1.1ex] 6 &amp; -1 \\end{pmatrix}+ 4 \\cdot \\begin{pmatrix} -1 &amp; 3 \\\\[1.1ex] 0 &amp; 4 \\end{pmatrix} = \\begin{pmatrix} 12 &amp; -8 \\\\[1.1ex] 24 &amp; -4 \\end{pmatrix}+\\begin{pmatrix} -4 &amp; 12 \\\\[1.1ex] 0 &amp; 16 \\end{pmatrix} = \\begin{pmatrix} \\bm{8} &amp; \\bm{4} \\\\[1.1ex] \\bm{24} &amp; \\bm{12} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"530\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Propriedade do elemento neutro:<\/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-244c951ff1cce8dc60f6d66a781c0580_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1 \\cdot A = A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"71\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, ao multiplicar uma matriz por 1, n\u00e3o modificamos 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-0ee2c0afd1bf2904722701caca883125_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1 \\cdot   \\begin{pmatrix} 5 &amp; -4 &amp; 0 \\\\[1.1ex] 1 &amp; 3 &amp; -3 \\\\[1.1ex] 2 &amp; 9 &amp; 4 \\end{pmatrix}=\\begin{pmatrix} \\bm{5} &amp; \\bm{-4} &amp; \\bm{0} \\\\[1.1ex] \\bm{1} &amp; \\bm{3} &amp; \\bm{-3} \\\\[1.1ex] \\bm{2} &amp; \\bm{9} &amp; \\bm{4} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"275\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Essas s\u00e3o todas as propriedades do produto de um escalar e de uma matriz, ent\u00e3o esse \u00e9 o fim deste artigo. Esperamos que tenha gostado e, acima de tudo, que tenha aprendido a resolver a multiplica\u00e7\u00e3o de n\u00fameros com matrizes.<\/p>\n<p> Por outro lado, outras opera\u00e7\u00f5es matriciais ligadas \u00e0 multiplica\u00e7\u00e3o, e que s\u00e3o muito \u00fateis, s\u00e3o pot\u00eancias. Deixamos aqui a p\u00e1gina onde voc\u00ea aprender\u00e1 o que \u00e9 e como resolver a <a href=\"https:\/\/mathority.org\/pt\/potencias-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos\/\">pot\u00eancia de uma matriz<\/a> , caso tenha curiosidade.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina veremos como multiplicar um n\u00famero por uma matriz. Voc\u00ea tamb\u00e9m tem exemplos que o ajudar\u00e3o a entend\u00ea-lo perfeitamente e exerc\u00edcios resolvidos para que voc\u00ea possa praticar. Voc\u00ea tamb\u00e9m encontrar\u00e1 todas as propriedades do produto de um escalar e de uma matriz. Como multiplicar um n\u00famero por uma matriz? Para multiplicar um n\u00famero por &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/multiplicacao-de-um-numero-por-uma-matriz-2x2-e-3x3-exemplos-e-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Multiplicando um n\u00famero por uma 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-279","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 um n\u00famero por uma matriz - 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-um-numero-por-uma-matriz-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=\"Multiplica\u00e7\u00e3o de um n\u00famero por uma matriz - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina veremos como multiplicar um n\u00famero por uma matriz. 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