{"id":276,"date":"2023-07-06T23:03:43","date_gmt":"2023-07-06T23:03:43","guid":{"rendered":"https:\/\/mathority.org\/pt\/exemplos-de-matriz-transposta-ou-transposta-e-exercicios-resolvidos\/"},"modified":"2023-07-06T23:03:43","modified_gmt":"2023-07-06T23:03:43","slug":"exemplos-de-matriz-transposta-ou-transposta-e-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/exemplos-de-matriz-transposta-ou-transposta-e-exercicios-resolvidos\/","title":{"rendered":"Transpor matriz (ou transpor)"},"content":{"rendered":"<p>Nesta p\u00e1gina veremos como calcular a <strong>matriz de transposi\u00e7\u00e3o (ou transposi\u00e7\u00e3o)<\/strong> . Voc\u00ea tamb\u00e9m ver\u00e1 exerc\u00edcios resolvidos para n\u00e3o ter d\u00favidas de como transpor uma matriz.<\/p>\n<h2 class=\"wp-block-heading\"> Como calcular a matriz transposta (ou transposi\u00e7\u00e3o)?<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> A <strong>matriz transposta<\/strong> , tamb\u00e9m chamada de matriz transposta, \u00e9 a matriz obtida pela <strong>transforma\u00e7\u00e3o de linhas em colunas<\/strong> . A matriz transposta \u00e9 representada colocando um \u201ct\u201d no canto superior direito da matriz (A <sup>t<\/sup> ).<\/p>\n<p> Por <strong>exemplo<\/strong> , vamos transpor a seguinte 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-8613db3e71f21d9ee2c4dc003600e32a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} 2 &amp; 3 &amp; 1 \\\\[1.1ex] 4 &amp; 5 &amp; 0   \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"120\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Para transpor a matriz A, basta <strong>trocar as linhas pelas colunas<\/strong> . Em outras palavras, 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:<\/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-e7cf9e274915aef7e44582556d188197_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^t= \\begin{pmatrix} 2 &amp; 4 \\\\[1.1ex] 3 &amp; 5 \\\\[1.1ex] 1 &amp; 0   \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"103\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Aqui est\u00e3o v\u00e1rios exemplos pr\u00e1ticos de como encontrar a matriz transposta:<\/p>\n<h2 class=\"wp-block-heading\"> Exemplos de matrizes transpostas<\/h2>\n<h3 class=\"wp-block-heading\"> Exemplo 1 <\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e3a8e6d458b2d60aabcedfe33c0297a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B= \\begin{pmatrix} 1 &amp; 5\\\\[1.1ex] 7 &amp; 2  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"96\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-321fc68b5d5d3c546461c29b80102a82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle B^t= \\begin{pmatrix} 1 &amp; 7\\\\[1.1ex] 5 &amp; 2  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"102\" 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\">Exemplo 2 <\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8295b1417a32fc9378584f87c67abc05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle C= \\begin{pmatrix} -1 &amp; 4 &amp; 3 \\\\[1.1ex] 5 &amp; 3 &amp; 2 \\\\[1.1ex] 6 &amp; 0 &amp; 9  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"137\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5274e4fc9bdc5939e3a5a08e6a8e41b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle C^t= \\begin{pmatrix} -1 &amp; 5 &amp; 6 \\\\[1.1ex] 4 &amp; 3 &amp; 0 \\\\[1.1ex] 3 &amp; 2 &amp; 9  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"143\" 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\"> Exemplo 3 <\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8e2d2268ed4a7fee24a06a7a7f7cd76b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle D= \\begin{pmatrix} 2 &amp; 6 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"126\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f1c2184833ec63a43162fde532f6e593_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle D^t= \\begin{pmatrix}2 \\\\[1.1ex] 6 \\\\[1.1ex] -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" 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\">Exemplo 4 <\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d83760eb84d950a0d31727e522d88f5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle E= \\begin{pmatrix} 9 &amp; 0  \\\\[1.1ex] 2 &amp; -1 \\\\[1.1ex] 5 &amp; 3   \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"112\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c01a100ebae58a267a286f88010a796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle E^t= \\begin{pmatrix} 9 &amp; 2 &amp; 5  \\\\[1.1ex] 0 &amp; -1 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"140\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Um dos usos da transposi\u00e7\u00e3o de matrizes \u00e9 <a href=\"https:\/\/mathority.org\/pt\/matriz-inversa\/\">calcular a matriz inversa com a f\u00f3rmula matricial anexada ou por determinantes<\/a> . Embora para utilizar este m\u00e9todo voc\u00ea tamb\u00e9m precise saber resolver determinantes, na p\u00e1gina vinculada voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o de todo o procedimento e tamb\u00e9m poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo.<\/p>\n<h2 class=\"wp-block-heading\"> Propriedades da matriz transposta<\/h2>\n<p> A matriz transposta possui as seguintes caracter\u00edsticas:<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Propriedade involucional:<\/span><\/strong> A transposta de uma matriz transposta \u00e9 igual \u00e0 matriz original.<\/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-a776188cca22c35019064e846b61b4b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(A^t\\right)^t = A\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"76\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Propriedade distributiva:<\/span><\/strong> adicionar duas matrizes e depois transpor o resultado equivale primeiro a transpor cada matriz e depois adicion\u00e1-las: <\/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-aff76a8cf7355ea147e7a885b034b462_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(A+B\\right)^t = A^t+B^t\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"152\" style=\"vertical-align: -5px;\"><\/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> <strong><span style=\"color:#1976d2;\">Propriedade linear (produto de matrizes):<\/span><\/strong> Multiplicar duas matrizes e depois transpor o resultado \u00e9 equivalente a primeiro transpor cada matriz e depois multiplic\u00e1-las, mas alternando sua ordem de multiplica\u00e7\u00e3o:<\/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-32a851b103c6805a1bf495dbdc04ddc1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(A\\cdot B\\right)^t = B^t\\cdot A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"134\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Propriedade linear (constante):<\/span><\/strong> Transpor o resultado do produto de uma matriz por uma constante equivale a multiplicar a matriz j\u00e1 transposta pela constante.<\/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-802c2bde61d7e419e73a5d5424661aec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(c\\cdot A\\right)^t = c\\cdot A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"114\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Matriz sim\u00e9trica:<\/span><\/strong> Se a transposta de uma matriz for igual \u00e0 matriz sem transposta, dizemos que \u00e9 uma <strong>matriz sim\u00e9trica:<\/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-d15c8f7ea597b852f92fdb47dbdb8c80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{pmatrix} 7 &amp; 1 &amp; 3 \\\\[1.1ex] 1 &amp; 4 &amp; 2 \\\\[1.1ex] 3 &amp; 2 &amp; 5  \\end{pmatrix} \\right.^t = \\begin{pmatrix} 7 &amp; 1 &amp; 3 \\\\[1.1ex] 1 &amp; 4 &amp; 2 \\\\[1.1ex] 3 &amp; 2 &amp; 5  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"202\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong><span style=\"color:#1976d2;\">Propriedade antissim\u00e9trica:<\/span><\/strong> Se ao transpor uma matriz matem\u00e1tica obtemos a mesma matriz mas com todos os elementos alterados de sinal, \u00e9 uma <strong>matriz antissim\u00e9trica:<\/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-5fcd7cfe60e5570bb668945b81540254_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{pmatrix} 0 &amp; 2 &amp; 4 \\\\[1.1ex] -2 &amp; 0 &amp; 6 \\\\[1.1ex] -4 &amp; -6 &amp; 0  \\end{pmatrix}\\right.^t = \\begin{pmatrix} 0 &amp; -2 &amp; -4 \\\\[1.1ex] 2 &amp; 0 &amp; -6 \\\\[1.1ex] 4 &amp; 6 &amp; 0  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"257\" style=\"vertical-align: 0px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina veremos como calcular a matriz de transposi\u00e7\u00e3o (ou transposi\u00e7\u00e3o) . Voc\u00ea tamb\u00e9m ver\u00e1 exerc\u00edcios resolvidos para n\u00e3o ter d\u00favidas de como transpor uma matriz. Como calcular a matriz transposta (ou transposi\u00e7\u00e3o)? A matriz transposta , tamb\u00e9m chamada de matriz transposta, \u00e9 a matriz obtida pela transforma\u00e7\u00e3o de linhas em colunas . A matriz &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-matriz-transposta-ou-transposta-e-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Transpor matriz (ou transpor)<\/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-276","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>Transpor matriz (ou transpor) - 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\/exemplos-de-matriz-transposta-ou-transposta-e-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Transpor matriz (ou transpor) - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina veremos como calcular a matriz de transposi\u00e7\u00e3o (ou transposi\u00e7\u00e3o) . Voc\u00ea tamb\u00e9m ver\u00e1 exerc\u00edcios resolvidos para n\u00e3o ter d\u00favidas de como transpor uma matriz. Como calcular a matriz transposta (ou transposi\u00e7\u00e3o)? A matriz transposta , tamb\u00e9m chamada de matriz transposta, \u00e9 a matriz obtida pela transforma\u00e7\u00e3o de linhas em colunas . 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