{"id":277,"date":"2023-07-06T22:49:54","date_gmt":"2023-07-06T22:49:54","guid":{"rendered":"https:\/\/mathority.org\/pt\/adicao-subtracao-de-matrizes-2x2-3x3-exemplos-exercicios-resolvidos\/"},"modified":"2023-07-06T22:49:54","modified_gmt":"2023-07-06T22:49:54","slug":"adicao-subtracao-de-matrizes-2x2-3x3-exemplos-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/adicao-subtracao-de-matrizes-2x2-3x3-exemplos-exercicios-resolvidos\/","title":{"rendered":"Como calcular adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes"},"content":{"rendered":"<p>Nesta p\u00e1gina veremos como <strong>adicionar e subtrair matrizes<\/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 de adi\u00e7\u00e3o de matrizes.<\/p>\n<h2 class=\"wp-block-heading\"> Como adicionar e subtrair matrizes?<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Para calcular uma <strong>adi\u00e7\u00e3o (ou subtra\u00e7\u00e3o) de duas matrizes,<\/strong> deve-se somar (ou subtrair) os elementos que ocupam a mesma posi\u00e7\u00e3o nas matrizes.<\/p>\n<h2 class=\"wp-block-heading\"> Exemplos: <\/h2>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/addition-et-soustraction-matricielle.webp\" alt=\"exemplos de adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes 2x2, opera\u00e7\u00f5es com matrizes\" class=\"wp-image-1267\" width=\"719\" height=\"373\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Observe que para adicionar ou subtrair duas matrizes, elas <strong>devem ter a mesma dimens\u00e3o.<\/strong> Por exemplo, as seguintes matrizes n\u00e3o podem ser adicionadas porque a primeira \u00e9 uma matriz 2&#215;2 e a segunda \u00e9 uma matriz 3&#215;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-082c648e15685c4ddeac2cc2da502d96_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 3 \\\\[1.1ex] 0 &amp; 2 \\end{pmatrix}  + \\begin{pmatrix} 5 &amp; 6 \\\\[1.1ex] -2 &amp; 4 \\\\[1.1ex] 7 &amp; 1 \\end{pmatrix} \\ \\longleftarrow \\ \\color{red}  \\bm{\\times}}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"247\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Exerc\u00edcios resolvidos para adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes<\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Calcule a seguinte soma de matrizes 2&#215;2: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-daddition-de-matrices-22.webp\" alt=\"exerc\u00edcio resolvido passo a passo para adi\u00e7\u00e3o de matrizes 2x2\" class=\"wp-image-1271\" width=\"175\" 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 soma de duas matrizes quadradas 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-1d9428ad89a6bd149d5e63bc500879ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2 &amp; 3 \\\\[1.1ex] 4 &amp; 1  \\end{pmatrix} + \\begin{pmatrix} 2 &amp; 1 \\\\[1.1ex] 3 &amp; -1  \\end{pmatrix} =  \\begin{pmatrix} 2+2 &amp; 3+1 \\\\[1.1ex] 4+3 &amp; 1+(-1)  \\end{pmatrix} = \\begin{pmatrix} \\bm{4} &amp; \\bm{4} \\\\[1.1ex] \\bm{7} &amp; \\bm{0}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"411\" 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> Execute a seguinte subtra\u00e7\u00e3o de matriz: <\/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\/soustraction-matricielle-resolue-exercice-32.webp\" alt=\"exerc\u00edcio resolvido subtra\u00e7\u00e3o passo a passo de matrizes, opera\u00e7\u00f5es com matrizes\" width=\"193\" height=\"99\" 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 subtra\u00e7\u00e3o de duas matrizes de dimens\u00e3o 3\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-c371e1f01df59f4b8abb018e476e66d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 5 &amp; 2  \\\\[1.1ex] 1 &amp; 6 \\\\[1.1ex] -3 &amp; 0  \\end{pmatrix} - \\begin{pmatrix} 4 &amp; 6 \\\\[1.1ex] -3 &amp; 1 \\\\[1.1ex]-2 &amp; 5 \\end{pmatrix} =  \\begin{pmatrix} 5-4 &amp; 2-6  \\\\[1.1ex] 1-(-3) &amp; 6-1 \\\\[1.1ex] -3-(-2) &amp; 0-5  \\end{pmatrix}  = \\begin{pmatrix} \\bm{1}&amp;  \\bm{-4} \\\\[1.1ex] \\bm{4} &amp; \\bm{5} \\\\[1.1ex] \\bm{-1} &amp; \\bm{-5} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"477\" 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> Encontre o resultado da seguinte soma de matrizes de dimens\u00e3o 3\u00d73: <\/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-daddition-de-matrices-33.webp\" alt=\"exerc\u00edcio resolvido passo a passo de adi\u00e7\u00e3o de matrizes 3x3, opera\u00e7\u00f5es com matrizes\" width=\"255\" height=\"102\" 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 soma de duas matrizes quadradas de ordem 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-280299cb0b37e1a585466c4570439ec4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 4 &amp; 1 &amp; -2 \\\\[1.1ex] 0 &amp; 3 &amp; 2 \\\\[1.1ex] 5 &amp; 1 &amp; 6 \\end{pmatrix} + \\begin{pmatrix} 2 &amp; 0 &amp; 5 \\\\[1.1ex] -3 &amp; 4 &amp; 1 \\\\[1.1ex] 1 &amp; 7 &amp; 8 \\end{pmatrix} =  \\begin{pmatrix} 4+2 &amp; 1+0 &amp; -2+5 \\\\[1.1ex] 0+(-3) &amp; 3+4 &amp; 2+1 \\\\[1.1ex] 5+1 &amp; 1+7 &amp; 6+8 \\end{pmatrix} = \\begin{pmatrix} \\bm{6}&amp;  \\bm{1} &amp; \\bm{3} \\\\[1.1ex] \\bm{-3} &amp; \\bm{7} &amp; \\bm{3} \\\\[1.1ex] \\bm{6} &amp; \\bm{8} &amp; \\bm{14} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"595\" 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:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 4<\/h3>\n<p> Calcule a seguinte adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes quadradas de ordem 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-combine-daddition-et-de-soustraction-de-matrices-22152.webp\" alt=\"exerc\u00edcio resolvido passo a passo adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes 2x2, opera\u00e7\u00f5es com matrizes\" width=\"323\" 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 opera\u00e7\u00e3o combinada com adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes quadradas 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-c9fa4dba7699c0035ce5081756b4f62e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 5 &amp; 1 \\\\[1.1ex] -2 &amp; 4  \\end{pmatrix} +  \\begin{pmatrix} 6 &amp; -2 \\\\[1.1ex] 3 &amp; -5  \\end{pmatrix} -\\begin{pmatrix} -3 &amp; 4 \\\\[1.1ex] 1 &amp; -2  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"276\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, primeiro adicionamos as matrizes \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-e1544e4da9d5ad2ea3ec2e4ad0326023_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 11 &amp; -1 \\\\[1.1ex] 1 &amp; -1  \\end{pmatrix}  -\\begin{pmatrix} -3 &amp; 4 \\\\[1.1ex] 1 &amp; -2  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"188\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E ent\u00e3o calculamos a subtra\u00e7\u00e3o de 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-bd7f32fc7c9429fdfc3b5b745e85975c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} \\bm{14} &amp; \\bm{-5} \\\\[1.1ex] \\bm{0} &amp; \\bm{1}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"77\" 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> Resolva a seguinte adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes: <\/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-combine-daddition-et-de-soustraction-de-matrices-33.webp\" alt=\"exerc\u00edcio resolvido passo a passo adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes 3x3, opera\u00e7\u00f5es com matrizes\" width=\"437\" 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 uma opera\u00e7\u00e3o combinada de subtra\u00e7\u00e3o e adi\u00e7\u00e3o de matrizes quadradas 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-ae66268adcd61258654056815542cf58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}5 &amp; 3 &amp; -1 \\\\[1.1ex] 6 &amp; -4 &amp; -2 \\\\[1.1ex] 2 &amp; 3 &amp; 2 \\end{pmatrix}-\\begin{pmatrix} 3 &amp; 2 &amp; 6 \\\\[1.1ex]-1 &amp; 5 &amp; 0 \\\\[1.1ex] 2 &amp; 4 &amp; 1 \\end{pmatrix} + \\begin{pmatrix}2 &amp; -1 &amp; 5 \\\\[1.1ex] -3 &amp; 1 &amp; 4 \\\\[1.1ex] 6 &amp; 0 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"373\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Primeiro, resolvemos a subtra\u00e7\u00e3o de 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-4401b28babce2beaaa6f840c4ed8c959_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}2 &amp; 1 &amp; -7 \\\\[1.1ex] 7 &amp; -9 &amp; -2 \\\\[1.1ex] 0 &amp; -1 &amp; 1 \\end{pmatrix}+\\begin{pmatrix}2 &amp; -1 &amp; 5 \\\\[1.1ex] -3 &amp; 1 &amp; 4 \\\\[1.1ex] 6 &amp; 0 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"247\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente adicionamos 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-ffba1ade3d98c434960b54fc0c7ffe1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} \\bm{4} &amp; \\bm{0} &amp; \\bm{-2} \\\\[1.1ex] \\bm{4} &amp; \\bm{-8} &amp; \\bm{2}  \\\\[1.1ex] \\bm{6} &amp; \\bm{-1} &amp; \\bm{4} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"108\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Agora que voc\u00ea sabe como somar e subtrair matrizes, \u00e9 um bom momento para ver como <a href=\"https:\/\/mathority.org\/pt\/multiplicacao-de-matrizes-2x2-e-3x3-exemplos-e-exercicios-resolvidos-passo-a-passo\/\">multiplicar matrizes<\/a> , certamente a mais importante das opera\u00e7\u00f5es com matrizes. Voc\u00ea tamb\u00e9m encontrar\u00e1 exerc\u00edcios passo a passo de multiplica\u00e7\u00e3o de matrizes resolvidos para praticar, como em todas as p\u00e1ginas deste site. \ud83d\ude09<\/p>\n<h2 class=\"wp-block-heading\"> Adicionar propriedades de matriz<\/h2>\n<p> A adi\u00e7\u00e3o de matrizes tem as seguintes caracter\u00edsticas:<\/p>\n<ul>\n<li> A adi\u00e7\u00e3o de matrizes tem a <strong><span style=\"color:#1976d2;\">propriedade comutativa<\/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-82f98b26399adb4b532b48c18bbbae16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A +B = B + A\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"122\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Portanto, a ordem em que somamos as matrizes \u00e9 a mesma. Para demonstrar isso, adicionaremos duas matrizes alterando sua ordem e voc\u00ea ver\u00e1 como o resultado \u00e9 o mesmo.<\/p>\n<p> Portanto, procedemos \u00e0 adi\u00e7\u00e3o de duas matrizes em uma determinada ordem:<\/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-a7eb454436dc3268ae8d6d2b62f395a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1 &amp; 3 \\\\[1.1ex] 2 &amp; -1 \\end{pmatrix}  +  \\begin{pmatrix} 4 &amp; 1 \\\\[1.1ex] 5 &amp; 2  \\end{pmatrix}= \\begin{pmatrix} \\bm{5} &amp; \\bm{4} \\\\[1.1ex] \\bm{7} &amp; \\bm{1}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"237\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Observe que se invertermos a ordem de adi\u00e7\u00e3o das matrizes, o resultado permanece o mesmo: <\/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-c1e9cd77bc490913ed30ff63815da355_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 4 &amp; 1 \\\\[1.1ex] 5 &amp; 2  \\end{pmatrix}  +  \\begin{pmatrix} 1 &amp; 3 \\\\[1.1ex] 2 &amp; -1 \\end{pmatrix}=  \\begin{pmatrix} \\bm{5} &amp; \\bm{4} \\\\[1.1ex] \\bm{7} &amp; \\bm{1}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"237\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<ul>\n<li> Outra propriedade da adi\u00e7\u00e3o de matrizes \u00e9 a do <strong style=\"color: rgb(25, 118, 210);\">elemento oposto:<\/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-5195a54259faa7e6f78b82f517a58e2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A + (-A) =0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Em outras palavras, se somarmos uma matriz mais a mesma matriz, mas com todos os seus elementos com sinais alterados, o resultado ser\u00e1 uma matriz zero:<\/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-add832e83fe554143cbd4c710315c1c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{pmatrix} 4 &amp; 1 &amp; -3 \\\\[1.1ex] 2 &amp; 0 &amp; 9 \\end{pmatrix} + \\begin{pmatrix} -4 &amp; -1 &amp; 3 \\\\[1.1ex] -2 &amp; 0 &amp; -9 \\end{pmatrix} =  \\begin{pmatrix} \\bm{0} &amp; \\bm{0} &amp; \\bm{0} \\\\[1.1ex] \\bm{0} &amp; \\bm{0} &amp; \\bm{0}  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"353\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> A adi\u00e7\u00e3o de matrizes tamb\u00e9m possui a <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-ac2f2c3b2989e505a4d61bab8759a13d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A + 0 =A\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"80\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Esta propriedade \u00e9 a mais \u00f3bvia, refere-se ao fato de que qualquer matriz mais uma matriz cheia de zeros equivale \u00e0 mesma 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-ac7b0ba246075c196188798be2c6a034_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2 &amp; 1 &amp; 5 \\\\[1.1ex] -3 &amp; 4 &amp; 9 \\\\[1.1ex] 1 &amp; 12 &amp; 6 \\end{pmatrix} + \\begin{pmatrix} 0 &amp; 0  &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; 0  \\end{pmatrix} =  \\begin{pmatrix} \\bm{2} &amp; \\bm{1} &amp; \\bm{5} \\\\[1.1ex] \\bm{-3} &amp; \\bm{4} &amp; \\bm{9} \\\\[1.1ex] \\bm{1} &amp; \\bm{12} &amp; \\bm{6} \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"351\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> A adi\u00e7\u00e3o de matrizes tem a <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-9b1aee88fd57af78c40429c93c7a2136_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\left( A + B \\right) + C  =A +  \\left(  B + C \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"219\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, a ordem em que somamos as matrizes \u00e9 a mesma. Veja o exemplo a seguir, onde somamos 3 matrizes com ordens diferentes e o resultado \u00e9 o mesmo: <\/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-bae8e10bca43351f3a84f83bfe50ab55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A =  \\begin{pmatrix} 2  \\\\[1.1ex] 1 \\end{pmatrix}  \\qquad B = \\begin{pmatrix} 4  \\\\[1.1ex] -1  \\end{pmatrix} \\qquad C = \\begin{pmatrix} 3  \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"310\" 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-2cc2b7a14cacc7e403cd729cd863d309_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\left( A + B \\right) + C &amp; =\\left(  \\begin{pmatrix} 2  \\\\[1.1ex] 1  \\end{pmatrix}   +  \\begin{pmatrix} 4  \\\\[1.1ex] -1  \\end{pmatrix} \\right) + \\begin{pmatrix} 3  \\\\[1.1ex] 0  \\end{pmatrix}  \\\\[2ex] &amp; =   \\begin{pmatrix} 6  \\\\[1.1ex] 0  \\end{pmatrix} + \\begin{pmatrix} 3  \\\\[1.1ex] 0 \\end{pmatrix} \\\\[2ex] &amp; =\\begin{pmatrix} \\bm{9}  \\\\[1.1ex] \\bm{0} \\end{pmatrix} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"204\" width=\"313\" 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-7ab1f88e74b139451eccb0471988c3db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} A +  \\left(  B + C \\right) &amp; = \\begin{pmatrix} 2  \\\\[1.1ex] 1  \\end{pmatrix}  + \\left( \\begin{pmatrix} 4  \\\\[1.1ex] -1  \\end{pmatrix}  +\\begin{pmatrix} 3  \\\\[1.1ex] 0  \\end{pmatrix} \\right) \\\\[2ex] &amp; =  \\begin{pmatrix} 2  \\\\[1.1ex] 1  \\end{pmatrix} + \\begin{pmatrix} 7  \\\\[1.1ex] -1  \\end{pmatrix} \\\\[2ex] &amp; = \\begin{pmatrix}  \\bm{9}  \\\\[1.1ex] \\bm{0}\\end{pmatrix} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"204\" width=\"314\" style=\"vertical-align: 0px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina veremos como adicionar e subtrair matrizes . 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 de adi\u00e7\u00e3o de matrizes. Como adicionar e subtrair matrizes? Para calcular uma adi\u00e7\u00e3o (ou subtra\u00e7\u00e3o) de duas matrizes, deve-se somar (ou subtrair) &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/adicao-subtracao-de-matrizes-2x2-3x3-exemplos-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Como calcular adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes<\/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-277","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>Como calcular adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/adicao-subtracao-de-matrizes-2x2-3x3-exemplos-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Como calcular adi\u00e7\u00e3o e subtra\u00e7\u00e3o de matrizes - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina veremos como adicionar e subtrair matrizes . 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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 de adi\u00e7\u00e3o de matrizes. Como adicionar e subtrair matrizes? 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