{"id":295,"date":"2023-07-06T17:07:12","date_gmt":"2023-07-06T17:07:12","guid":{"rendered":"https:\/\/mathority.org\/pt\/extensao-de-uma-matriz-em-funcao-de-um-parametro-exemplos-e-exercicios-resolvidos-de-matrizes-2x2-3x3-3x4-4x4\/"},"modified":"2023-07-06T17:07:12","modified_gmt":"2023-07-06T17:07:12","slug":"extensao-de-uma-matriz-em-funcao-de-um-parametro-exemplos-e-exercicios-resolvidos-de-matrizes-2x2-3x3-3x4-4x4","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/extensao-de-uma-matriz-em-funcao-de-um-parametro-exemplos-e-exercicios-resolvidos-de-matrizes-2x2-3x3-3x4-4x4\/","title":{"rendered":"Intervalo de um array baseado em um par\u00e2metro"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 como calcular a <strong>classifica\u00e7\u00e3o de uma tabela com base em um par\u00e2metro.<\/strong> Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos passo a passo e exerc\u00edcios resolvidos sobre como encontrar o contradom\u00ednio de uma matriz com base em um par\u00e2metro.<\/p>\n<p> Para compreender totalmente o procedimento de estudo do posto de matrizes com par\u00e2metros, \u00e9 importante que voc\u00ea j\u00e1 saiba <a href=\"https:\/\/mathority.org\/pt\">calcular o posto de uma matriz por determinantes<\/a> . Portanto, recomendamos que voc\u00ea aprenda essas duas coisas antes de continuar lendo.<\/p>\n<h2 class=\"wp-block-heading\"> Como calcular o intervalo de um array com base em um par\u00e2metro. Exemplo:<\/h2>\n<ul>\n<li> Determina o intervalo da matriz A com base em diferentes valores de par\u00e2metros\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/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-0aa5688f2845a0225149f448466c943c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\begin{pmatrix} a+1 &amp; -1 &amp; a+1 \\\\[1.1ex] 0 &amp; -1 &amp; 0   \\\\[1.1ex] 1 &amp; -2 &amp; a  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"198\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A matriz A ter\u00e1 no m\u00e1ximo classifica\u00e7\u00e3o 3, porque \u00e9 uma matriz de ordem 3. Portanto, a primeira coisa que precisamos fazer \u00e9 <strong>resolver o determinante de toda a matriz 3&#215;3<\/strong> com <a href=\"https:\/\/mathority.org\/pt\/determinantes-3x3-exemplos-de-regras-sarrus-e-exercicios-resolvidos\/\">a regra de Sarrus<\/a> , para ver se ele pode ser de posto 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-835a881061438326519f4660b4c394fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} a+1 &amp; -1 &amp; a+1 \\\\[1.1ex] 0 &amp; -1 &amp; 0   \\\\[1.1ex] 1 &amp; -2 &amp; a  \\end{vmatrix} &amp; =-a(a+1)+0+0+a+1-0-0 \\\\ &amp; =-a^2-a+a+1  \\\\[1.5ex] &amp; =-a^2+1 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"429\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O resultado do determinante \u00e9 uma fun\u00e7\u00e3o do par\u00e2metro<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> . <strong>Portanto, definimos o resultado igual a 0<\/strong> para ver quando a tabela ser\u00e1 de classifica\u00e7\u00e3o 2 e quando ser\u00e1 de classifica\u00e7\u00e3o 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-7cf08fe725290ac099f54916fa4c5dcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -a^2+1 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> E resolvemos a equa\u00e7\u00e3o 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-18b6f04242243eeefa0cd5892b29f4d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a^2 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" 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-ad7c0d92bbec913193a85949c7a0bfa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\sqrt{a^2} = \\sqrt{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -1px;\"><\/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-1191c881d84f673236382966b4e709ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{a = \\pm 1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, quando<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> seja +1 ou -1, o determinante 3\u00d73 ser\u00e1 0 e, portanto, o posto da matriz n\u00e3o ser\u00e1 3. Por outro lado, quando<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> for diferente de +1 e -1, o determinante ser\u00e1 diferente de 0 e, portanto, a matriz ter\u00e1 posto 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-60e6f80d73c96b28458d7790d98d0a5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c}  \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq +1,-1 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"354\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Agora vamos ver o que acontece quando<\/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-bbdf9897658213f9f2ad0b6a3d8d87cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{a=+1} :\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"65\" style=\"vertical-align: -2px;\"><\/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-53dde6f61dc01cac5c0a0705c44a7433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = +1 \\longrightarrow A= \\begin{pmatrix} 2 &amp; -1 &amp; 2 \\\\[1.1ex] 0 &amp; -1 &amp; 0   \\\\[1.1ex] 1 &amp; -2 &amp; 1  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"230\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como vimos anteriormente, quando<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 1, o determinante da matriz \u00e9 0. Portanto, n\u00e3o pode ser de posto 3. Tentamos agora calcular um <a href=\"https:\/\/mathority.org\/pt\/determinantes-2x2-exemplos-e-exercicios-resolvidos\/\">determinante 2\u00d72<\/a> diferente de 0 dentro da matriz, por exemplo, o do canto superior esquerdo:<\/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-d291f322f9d3f392e46568817e531a84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle   \\begin{vmatrix} 2 &amp; -1 \\\\[1.1ex] 0 &amp; -1 \\end{vmatrix} =-2-0= -2 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O determinante de ordem 2 \u00e9 diferente de 0. Assim, quando o par\u00e2metro<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> ou +1, a <strong>classifica\u00e7\u00e3o da matriz ser\u00e1 2:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d00c47041db87183749744eaf6789fd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c}  \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = +1 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Uma vez que vemos o contradom\u00ednio da matriz quando<\/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-910ad8735da02f7dffe9cd0fda341d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a \\neq +1,-1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\"><\/p>\n<p> e quando<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-10f3012b6955e51b81c57a6e2e57b7df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=+1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"55\" style=\"vertical-align: -2px;\"><\/p>\n<p> Vamos ver o que acontece quando<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d04c75a36ec68cca9920060cc558b99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{a = -1} :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" 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-f723d9c6b9f786b8c405ac7ec2d8bf1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = -1 \\longrightarrow A=  \\begin{pmatrix} 0 &amp; -1 &amp; 0 \\\\[1.1ex] 0 &amp; -1 &amp; 0   \\\\[1.1ex] 1 &amp; -2 &amp; -1  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como vimos no in\u00edcio, quando<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> es -1 e o determinante da matriz \u00e9 0. Portanto, ela n\u00e3o pode ser definida como classifica\u00e7\u00e3o 3. Portanto, devemos tentar encontrar um determinante de 2\u00d72 na matriz que seja diferente de 0, por exemplo, o menor parte da matriz. 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-cdc9bd6d9ad083e1e38f53079aebb5e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle   \\begin{vmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; -2  \\end{vmatrix} = 0-(-1)= 1\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O determinante da dimens\u00e3o 2 \u00e9 diferente de 0. Assim, quando o par\u00e2metro<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> ou -1, a <strong>classifica\u00e7\u00e3o da tabela ser\u00e1 2:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d12346bae2f327e7e1ee6c5276a599cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = -1 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Encontramos, portanto, 3 casos diferentes em que a classifica\u00e7\u00e3o da matriz A depende do valor que o par\u00e2metro assume<\/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-4ae924c776e55c0f2987a783307cd9fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> Aqui est\u00e1 o <strong>resumo<\/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-dc3a7ebea32c871ab7971a276decc60a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq +1,-1 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = +1 \\ \\longrightarrow \\ Rg(A)=2} \\\\[3ex]  \\color{black} \\bm{a = -1 \\ \\longrightarrow \\ Rg(A)=2} \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"354\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Agora que voc\u00ea sabe como discutir o intervalo de matrizes dependentes de par\u00e2metros, pode praticar os exerc\u00edcios passo a passo abaixo. Para resolv\u00ea-los, as <a href=\"https:\/\/mathority.org\/pt\/propriedades-de-determinantes-exemplos-e-exercicios-2x2-3x3\/\">propriedades dos determinantes<\/a> certamente ir\u00e3o ajud\u00e1-lo, ent\u00e3o se voc\u00ea n\u00e3o tem muita clareza sobre eles, aconselho primeiro a dar uma olhada na p\u00e1gina do link, onde cada um deles \u00e9 explicado com exemplos.<\/p>\n<h2 class=\"wp-block-heading\"> Corrigidos problemas de intervalo de matriz baseado em par\u00e2metros<\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Estude o intervalo da tabela a seguir com base no valor do par\u00e2metro <\/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-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7f53b08bcf2e2660dbb7c0aeb6fd369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; 1 &amp; a \\\\[1.1ex] 2 &amp; 2 &amp; -4 \\\\[1.1ex] 2 &amp; 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"136\" 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\"> A matriz A ter\u00e1 no m\u00e1ximo classifica\u00e7\u00e3o 3, porque \u00e9 uma matriz 3\u00d73. Portanto, a primeira coisa que precisamos fazer \u00e9 resolver o determinante de toda a matriz (com a regra de Sarrus), para ver se ela pode ser de posto 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-d2539698cbcf9f06d2890d17da76174f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; 1 &amp; a \\\\[1.1ex] 2 &amp; 2 &amp; -4 \\\\[1.1ex] 2 &amp; 1 &amp; 0 \\end{vmatrix}  =0-8+2a-4a+12-0 =-2a+4\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"382\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Definimos o resultado igual a 0 para ver quando o array ter\u00e1 a classifica\u00e7\u00e3o 2 e quando a classifica\u00e7\u00e3o 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-7042ae953fdcbe91d08fa963be26f7c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2a+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"94\" style=\"vertical-align: -2px;\"><\/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-90183d93145fd04e7a774c8a72bc3f1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2a=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"78\" 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-21d4999dede651fdb38c5b047b8e805d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a=\\cfrac{-4}{-2} = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"97\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, quando<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> for diferente de 2, o determinante 3\u00d73 ser\u00e1 diferente de 0 e, portanto, o posto da matriz ser\u00e1 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-a4c4e0bfd1194afe82d8807c033e7551_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq 2 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"310\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos ver o que acontece quando <\/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-f2abbabd80372bf9bc248f12cebd5fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=2 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" 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-c131e19dd5d5c0d7826306103b4e118b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = 2 \\longrightarrow A= \\begin{pmatrix} 3 &amp; 1 &amp; 2 \\\\[1.1ex] 2 &amp; 2 &amp; -4 \\\\[1.1ex] 2 &amp; 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"216\" 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-b97f01989b5e9679f95d300cd64f3735_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} 3 &amp; 1 &amp; 2 \\\\[1.1ex] 2 &amp; 2 &amp; -4 \\\\[1.1ex] 2 &amp; 1 &amp; 0 \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"164\" 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-77b38ebf03b8ed059edefd523c5ca1f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; 1  \\\\[1.1ex] 2 &amp; 2 \\end{vmatrix} = 6-2 = 4 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" 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-1f174f72890bce94d148e1f6e88681ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = 2 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"310\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Encontramos portanto 2 casos em que a imagem da matriz A varia com o valor que o par\u00e2metro assume: <\/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-41e5cc7b6e9b3204f26e1c64e46f7057_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq 2 \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = 2\\ \\longrightarrow \\ Rg(A)=2}  \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"122\" width=\"304\" 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 intervalo da tabela a seguir com base no valor do par\u00e2metro <\/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-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5b28f21cc2e7211d9dae9b6685b541fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] a &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; a \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"150\" 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\"> A matriz A ter\u00e1 no m\u00e1ximo classifica\u00e7\u00e3o 3, porque \u00e9 uma matriz 3\u00d73. Portanto, a primeira coisa que precisamos fazer \u00e9 resolver o determinante de toda a matriz (com a regra de Sarrus), para ver se ela pode ser de posto 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-30c8c16fea09001059a5d66727fc7be3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] a &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; a \\end{vmatrix} &amp; =2a-12-2a+2+12-2a^2 \\\\ &amp;=2-2a^2\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"335\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Definimos o resultado igual a 0 para ver quando o array ter\u00e1 a classifica\u00e7\u00e3o 2 e quando a classifica\u00e7\u00e3o 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-28c4eeb004bd0bf3db692ee22c659a40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2-2a^2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"89\" 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-e39820ff30d5df06ac09f254dcebeef0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2a^2=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" 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-3c5cad133a274f40a2151ad9e9310825_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a^2=\\cfrac{-2}{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"74\" style=\"vertical-align: -12px;\"><\/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-8b5cd6314cc67aa83d49e16072e9314b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a^2=1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" 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-63f049dc27947cfc24afdd331acefe23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a=\\pm 1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, quando<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> for diferente de +1 e -1, o determinante 3\u00d73 ser\u00e1 diferente de 0 e, portanto, o posto da matriz ser\u00e1 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-e7d26d825cd80ee861dd13168dafd408_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq +1, -1 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"354\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos ver o que acontece quando <\/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-ee9005a2708f5bcb0f0fba0cefed3dfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=+1 :\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"65\" style=\"vertical-align: -2px;\"><\/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-7b95d408f076c4978c8605380a277cdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = +1 \\longrightarrow A= \\begin{pmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] 1 &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"244\" 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-fcd50b9549925b5011a6c20943c326ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] 1 &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; 1 \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"178\" 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-03242296f208e07b9c4d634f0b7724cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}  2 &amp; 1 \\\\[1.1ex]  1 &amp; 3 \\end{vmatrix} = 6-1 = 5 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" 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-2550b439990981d1b74f72b1649a57e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = +1 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos ver o que acontece quando <\/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-53cc36b0e502c4e9a0aa575015035a8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=-1 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" 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-a2d16421400df26760d811229215ac83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = -1 \\longrightarrow A= \\begin{pmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] -1 &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"258\" 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-ecd0b86cd6c59a0911f0c39ca7599806_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] -1 &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; -1  \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"191\" 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-378e43f0ef61ccabf82dacb5ac70466f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 2 &amp; 2  \\\\[1.1ex] -1 &amp; 1 \\end{vmatrix} =2-(-2) = 4 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"213\" 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-d12346bae2f327e7e1ee6c5276a599cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = -1 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Encontramos, portanto, 3 casos em que o intervalo da matriz A varia dependendo do valor que o par\u00e2metro assume: <\/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-6bf1904cee51914e041d94f588fed84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq +1,-1 \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = +1\\ \\longrightarrow \\ Rg(A)=2} \\\\[3ex] \\color{black} \\bm{a = -1\\ \\longrightarrow \\ Rg(A)=2}  \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"348\" 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 3<\/h3>\n<p> Calcula o intervalo da tabela a seguir com base no valor do par\u00e2metro <\/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-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-090a99d3b4111785433e5c769589eb01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} a+1 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; a-3  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"184\" 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\"> A matriz A ter\u00e1 no m\u00e1ximo classifica\u00e7\u00e3o 3, porque \u00e9 uma matriz 3\u00d73. Portanto, a primeira coisa que precisamos fazer \u00e9 resolver o determinante de toda a matriz (com a regra de Sarrus), para ver se ela pode ser de posto 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-fec1cb52bb87fa2bccb40b70e1f21c7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} a+1 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; a-3 \\end{vmatrix} &amp; =(a+1)(a-3) +2+0-5+6(a+1)-0 \\\\ &amp; = a^2-3a+a-3 +2-5+6a+6 \\\\[1.5ex] &amp; =a^2+4a\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"468\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Definimos o resultado igual a 0 para ver quando o array ter\u00e1 a classifica\u00e7\u00e3o 2 e quando a classifica\u00e7\u00e3o 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-f8e26b9f10414656086a0c25d28ea04f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a^2+4a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Esta \u00e9 uma equa\u00e7\u00e3o quadr\u00e1tica incompleta, ent\u00e3o extra\u00edmos um fator comum:<\/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-3f6035239798b59504a776dac1f0e21a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a(a+4)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E definimos cada termo igual a 0:<\/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-43b38da320da538e46c6b4515de48568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a(a+4)=0 \\longrightarrow \\begin{cases} \\bm{a = 0} \\\\[2ex] a+4=0  \\ \\longrightarrow \\ \\bm{a=-4}\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"328\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Obtivemos 0 e -4 como solu\u00e7\u00f5es. Portanto, quando<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> for diferente de 0 e -4, o determinante 3\u00d73 ser\u00e1 diferente de 0 e, portanto, o posto da matriz ser\u00e1 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-15908960ef2cfcd2105c4b901fb6cb49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq 0, -4 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"340\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos ver o que acontece quando <\/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-d229e6228a70e103acbec8ca88c12d7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=0 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" 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-d97b25f01cb00d4677da0de5b4340ddb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = 0 \\longrightarrow A= \\begin{pmatrix} 1 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; -3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"230\" 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-9e0f3c315588dff8274873001f727a69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} 1 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; -3 \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"178\" 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-c132b22650c707d9f410c3d9c1e8da35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 1  \\\\[1.1ex] 0 &amp; 1 \\end{vmatrix} = 1-0 = 1 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" 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-d33aeec452b54112a958bfeadf014fe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = 0 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"310\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos ver o que acontece quando <\/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-0287c5c8b769f316fb7d382ea3332fa7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=-4 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" 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-9ce7e40d9d78ecddc5ee81fc799c8767_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = -4 \\longrightarrow A= \\begin{pmatrix} -3 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; -7  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"244\" 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-52d15d0bceeb1dbbc415fb4825ce9a05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} -3 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; -7 \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"178\" 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-d45971f1dbda32405246de38bb68bd92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} -3 &amp; 1 \\\\[1.1ex] 0 &amp; 1\\end{vmatrix} =-3-0 = -3 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"213\" 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-0551d4c8535193e378fc38c2e5580157_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = -4 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Encontramos, portanto, 3 casos em que o intervalo da matriz A varia dependendo do valor que o par\u00e2metro assume: <\/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-844f152985a2d84be1456501dfdc16e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq 0,-4 \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = 0\\ \\longrightarrow \\ Rg(A)=2} \\\\[3ex] \\color{black} \\bm{a = -4\\ \\longrightarrow \\ Rg(A)=2}  \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"334\" 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> Encontre a extens\u00e3o da seguinte matriz de dimens\u00e3o 3\u00d74 de acordo com o valor do par\u00e2metro <\/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-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4da7907bd0e8f80006ea47d2437b3f3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} -1&amp;-3&amp;-2&amp;1\\\\[1.1ex] 4&amp;12&amp;8&amp;-4\\\\[1.1ex] 2&amp;6&amp;4&amp;a \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"203\" 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\"> A matriz A ter\u00e1 no m\u00e1ximo posto 3, pois n\u00e3o podemos calcular nenhum <a href=\"https:\/\/mathority.org\/pt\/determinantes-4x4-por-exemplos-complementares-e-exercicios-resolvidos\/\">determinante 4\u00d74<\/a> . Portanto, a primeira coisa que precisamos fazer \u00e9 resolver todos os poss\u00edveis determinantes de ordem 3 (com a regra de Sarrus), para ver se podem ser 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-c2db025b8ecf4323d4a912d84a215d8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} -1&amp;-3&amp;-2\\\\[1.1ex] 4&amp;12&amp;8\\\\[1.1ex] 2&amp;6&amp;4 \\end{vmatrix} &amp; =-48-48-48+48+48+48 =\\bm{0}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"395\" 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-e40592bf6f8bfd13cb68a1fd0393cebb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} -1&amp;-3&amp;1\\\\[1.1ex] 4&amp;12&amp;-4\\\\[1.1ex] 2&amp;6&amp;a \\end{vmatrix} &amp; =-12a+24+24-24-24+12a=\\bm{0}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"414\" 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-ce1c28ae4120f0b37059b763e576d2eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} -1&amp;-2&amp;1\\\\[1.1ex] 4&amp;8&amp;-4\\\\[1.1ex] 2&amp;4&amp;a \\end{vmatrix} &amp; =-8a+16+16-16-16+8a=\\bm{0}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"396\" 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-668e9096b00b90ee4cc48d272b17e7bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} -3&amp;-2&amp;1\\\\[1.1ex] 12&amp;8&amp;-4\\\\[1.1ex] 6&amp;4&amp;a \\end{vmatrix} &amp; =-24a+48+48-48-48+24a=\\bm{0}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"414\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os resultados de todos os determinantes poss\u00edveis de ordem 3 s\u00e3o 0, qualquer que seja o valor 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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Assim, a matriz nunca ser\u00e1 de posto 3, pois n\u00e3o importa o valor que ela assuma<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> que nunca haver\u00e1 um determinante 3\u00d73 diferente de 0.<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o agora tentamos determinantes de dimens\u00e3o 2 \u00d7 2. No entanto, todos os determinantes de ordem 2 tamb\u00e9m d\u00e3o 0, exceto os seguintes:<\/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-c4408f1ccf562196943209356e50e892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} 8&amp;-4\\\\[1.1ex] 4&amp;a \\end{vmatrix} &amp; =8a+16 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"139\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora definimos o resultado igual a 0 e resolvemos a equa\u00e7\u00e3o: <\/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-f5494bb524be48bc22a1cb054556c3a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 8a+16=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"90\" style=\"vertical-align: -2px;\"><\/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-8a6fc020bc84c4ba3f1989065a2207fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 8a=-16\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" 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-02680bced4f2a76a7d23c5b9e6a2ecbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a=\\cfrac{-16}{8} =-2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"120\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, quando<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> for diferente de -2, o determinante 2\u00d72 ser\u00e1 diferente de 0 e, portanto, o posto da matriz ser\u00e1 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-04b3447f6e823c3e11b66919654e7a5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq -2 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos ver o que acontece quando <\/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-0e72cd3ad115f5d34fb5077b4d7d278a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=-2 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" 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-6ecbf63b188b46c05e67741cee83d7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = -2 \\longrightarrow A= \\begin{pmatrix} -1&amp;-3&amp;-2&amp;1\\\\[1.1ex] 4&amp;12&amp;8&amp;-4\\\\[1.1ex] 2&amp;6&amp;4&amp;-2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"297\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como vimos anteriormente, quando<\/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-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 -2, todos os determinantes de ordem 2 s\u00e3o 0. N\u00e3o pode, portanto, ser de posto 2. E como existe pelo menos um determinante 1\u00d71 diferente de 0, neste caso o posto da matriz \u00e9 1:<\/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-eb1cee57ae9619b3e4fdbf2357893425_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = -2 \\ \\longrightarrow \\ Rg(A)=1} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Encontramos portanto 2 casos em que a imagem da matriz A varia com o valor que o par\u00e2metro assume: <\/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-2bdfb67894431a4a08a3e791dcda0313_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq -2 \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = -2\\ \\longrightarrow \\ Rg(A)=1}   \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"122\" width=\"317\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 como calcular a classifica\u00e7\u00e3o de uma tabela com base em um par\u00e2metro. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos passo a passo e exerc\u00edcios resolvidos sobre como encontrar o contradom\u00ednio de uma matriz com base em um par\u00e2metro. Para compreender totalmente o procedimento de estudo do posto de matrizes com par\u00e2metros, \u00e9 importante que &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/extensao-de-uma-matriz-em-funcao-de-um-parametro-exemplos-e-exercicios-resolvidos-de-matrizes-2x2-3x3-3x4-4x4\/\"> <span class=\"screen-reader-text\">Intervalo de um array baseado em um par\u00e2metro<\/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":[9],"tags":[],"class_list":["post-295","post","type-post","status-publish","format-standard","hentry","category-calculadoras"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Faixa de uma matriz baseada em um par\u00e2metro -<\/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\/extensao-de-uma-matriz-em-funcao-de-um-parametro-exemplos-e-exercicios-resolvidos-de-matrizes-2x2-3x3-3x4-4x4\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Faixa de uma matriz baseada em um par\u00e2metro -\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 como calcular a classifica\u00e7\u00e3o de uma tabela com base em um par\u00e2metro. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos passo a passo e exerc\u00edcios resolvidos sobre como encontrar o contradom\u00ednio de uma matriz com base em um par\u00e2metro. Para compreender totalmente o procedimento de estudo do posto de matrizes com par\u00e2metros, \u00e9 importante que &hellip; Intervalo de um array baseado em um par\u00e2metro Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/extensao-de-uma-matriz-em-funcao-de-um-parametro-exemplos-e-exercicios-resolvidos-de-matrizes-2x2-3x3-3x4-4x4\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T17:07:12+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 minutos\" 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matriz baseada em um par\u00e2metro -","og_description":"Nesta p\u00e1gina voc\u00ea ver\u00e1 como calcular a classifica\u00e7\u00e3o de uma tabela com base em um par\u00e2metro. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos passo a passo e exerc\u00edcios resolvidos sobre como encontrar o contradom\u00ednio de uma matriz com base em um par\u00e2metro. Para compreender totalmente o procedimento de estudo do posto de matrizes com par\u00e2metros, \u00e9 importante que &hellip; Intervalo de um array baseado em um par\u00e2metro Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/extensao-de-uma-matriz-em-funcao-de-um-parametro-exemplos-e-exercicios-resolvidos-de-matrizes-2x2-3x3-3x4-4x4\/","article_published_time":"2023-07-06T17:07:12+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"5 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