{"id":293,"date":"2023-07-06T17:31:42","date_gmt":"2023-07-06T17:31:42","guid":{"rendered":"https:\/\/mathority.org\/pt\/classificacao-de-uma-matriz\/"},"modified":"2023-07-06T17:31:42","modified_gmt":"2023-07-06T17:31:42","slug":"classificacao-de-uma-matriz","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/classificacao-de-uma-matriz\/","title":{"rendered":"Calcule a classifica\u00e7\u00e3o de uma matriz por determinantes"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 e como calcular a <strong>imagem de uma matriz<\/strong> por determinantes. Al\u00e9m disso, voc\u00ea encontrar\u00e1 exemplos e exerc\u00edcios resolvidos para aprender como encontrar facilmente a extens\u00e3o de uma matriz. Al\u00e9m disso, voc\u00ea tamb\u00e9m ver\u00e1 as propriedades de intervalo de uma matriz.<\/p>\n<h2 class=\"wp-block-heading\"> Qual \u00e9 a classifica\u00e7\u00e3o de uma matriz?<\/h2>\n<p> A defini\u00e7\u00e3o de intervalo de uma matriz \u00e9:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> A <strong>classifica\u00e7\u00e3o de uma matriz<\/strong> \u00e9 a ordem da maior submatriz quadrada cujo determinante \u00e9 diferente de 0.<\/p>\n<p> Nesta p\u00e1gina aprenderemos sobre o contradom\u00ednio de uma matriz pelo m\u00e9todo dos determinantes, mas o contradom\u00ednio de uma matriz tamb\u00e9m pode ser determinado pelo m\u00e9todo gaussiano, embora seja mais lento e complicado.<\/p>\n<p> Depois de sabermos qual \u00e9 o contradom\u00ednio de uma matriz, veremos como determinar o contradom\u00ednio de uma matriz por determinantes. Mas tenha em mente que para resolver a extens\u00e3o de uma matriz, primeiro voc\u00ea precisa saber como calcular <a href=\"https:\/\/mathority.org\/pt\/determinantes-3x3-exemplos-de-regras-sarrus-e-exercicios-resolvidos\/\">determinantes 3&#215;3<\/a> .<\/p>\n<h2 class=\"wp-block-heading\"> Como saber a extens\u00e3o de uma matriz? Exemplo:<\/h2>\n<ul>\n<li> Calcule a extens\u00e3o da seguinte matriz de dimens\u00e3o 3\u00d74: <\/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-79e80ea42079a394262a4fcce5a863f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{cccc} 1 &amp; 3 &amp; 4 &amp; -1 \\\\[1.1ex] 0 &amp; 2 &amp; 1 &amp; -1  \\\\[1.1ex] 3 &amp; -1 &amp; 7 &amp; 2 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"191\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Sempre come\u00e7aremos tentando ver se a matriz tem classifica\u00e7\u00e3o m\u00e1xima resolvendo o maior determinante de ordem. E, se o determinante desta ordem for igual a 0, continuaremos a testar determinantes de ordem inferior at\u00e9 encontrarmos um diferente de 0.<\/p>\n<p> Neste caso, \u00e9 uma matriz de dimens\u00e3o 3\u00d74. <strong>Ser\u00e1, portanto, no m\u00e1ximo de posto 3<\/strong> , j\u00e1 que n\u00e3o podemos fazer nenhum determinante de ordem 4. Ent\u00e3o pegamos qualquer submatriz 3\u00d73 e vemos se seu determinante \u00e9 0. Por exemplo, resolvemos o determinante das 3 primeiras colunas com a regra de Sarrus:<\/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-819aaaa272025ce70b7852d00680483d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; \\cellcolor[HTML]{ABEBC6}3 &amp; \\cellcolor[HTML]{ABEBC6}4  &amp; -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp; \\cellcolor[HTML]{ABEBC6}2 &amp; \\cellcolor[HTML]{ABEBC6} 1 &amp; -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\\\[-2ex]\\cellcolor[HTML]{ABEBC6} 3 &amp;\\cellcolor[HTML]{ABEBC6} -1 &amp; \\cellcolor[HTML]{ABEBC6} 7 &amp; 2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"570\" 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-aedcd597b0cd9cd0ad11ab1d99bd0e5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 3 &amp; 4 \\\\[1.1ex] 0 &amp; 2 &amp; 1   \\\\[1.1ex] 3 &amp; -1 &amp; 7  \\end{vmatrix} = 14 + 9 + 0 - 24 + 1 - 0 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"318\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O determinante das colunas 1, 2 e 3 \u00e9 0. Devemos agora tentar outro determinante, por exemplo o das colunas 1, 2 e 4:<\/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-ddfbcde7994d5665983fda2423c82de3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; \\cellcolor[HTML]{ABEBC6} 3 &amp; 4  &amp; \\cellcolor[HTML]{ABEBC6}-1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp; &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp;\\cellcolor[HTML]{ABEBC6}2 &amp; 1 &amp; \\cellcolor[HTML]{ABEBC6} -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6}&amp; &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex]\\cellcolor[HTML]{ABEBC6} 3 &amp; \\cellcolor[HTML]{ABEBC6}-1 &amp; 7 &amp; \\cellcolor[HTML]{ABEBC6}2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"565\" 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-6f13263d4697369ed7d98bf7f972d15f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 3 &amp; -1 \\\\[1.1ex] 0 &amp; 2 &amp; -1   \\\\[1.1ex] 3 &amp; -1 &amp; 2  \\end{vmatrix} = 4 -9 + 0 + 6-1 - 0 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"314\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Tamb\u00e9m nos deu 0. Continuamos portanto a testar os determinantes de ordem 3 para ver se existem outros al\u00e9m de 0. Testamos agora o determinante formado pelas colunas 1, 3 e 4:<\/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-2682212fc905820bb8c2c2b73eeb49e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; 3 &amp; \\cellcolor[HTML]{ABEBC6}4  &amp; \\cellcolor[HTML]{ABEBC6}-1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp;2 &amp; \\cellcolor[HTML]{ABEBC6} 1 &amp; \\cellcolor[HTML]{ABEBC6} -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp;  &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex]\\cellcolor[HTML]{ABEBC6} 3 &amp;  -1 &amp; \\cellcolor[HTML]{ABEBC6} 7 &amp; \\cellcolor[HTML]{ABEBC6}2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"570\" 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-c84fbf30f1005e0bdd6496369c68efb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; 4 &amp; -1 \\\\[1.1ex] 0 &amp; 1 &amp; -1   \\\\[1.1ex] 3 &amp; 7 &amp; 2  \\end{vmatrix} = 2 -12+0 +3 +7- 0 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"309\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dos determinantes de ordem 3, basta tentar o determinante composto pelas colunas 2, 3 e 4:<\/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-610e7befed3409c44ad1b84a6c84605d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}1 &amp; \\cellcolor[HTML]{ABEBC6}3 &amp; \\cellcolor[HTML]{ABEBC6}4  &amp; \\cellcolor[HTML]{ABEBC6}-1 \\\\  &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex] 0 &amp; \\cellcolor[HTML]{ABEBC6}2 &amp; \\cellcolor[HTML]{ABEBC6} 1 &amp; \\cellcolor[HTML]{ABEBC6} -1 \\\\  &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex] 3 &amp; \\cellcolor[HTML]{ABEBC6} -1 &amp; \\cellcolor[HTML]{ABEBC6} 7 &amp; \\cellcolor[HTML]{ABEBC6}2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"570\" 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-6377c641072d9eba07fd2b9670ffbf50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle   \\begin{vmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex]  2 &amp; 1 &amp; -1  \\\\[1.1ex] -1 &amp; 7 &amp; 2 \\end{vmatrix} = 6+4-14-1+21-16 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"341\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> J\u00e1 tentamos todos os determinantes 3&#215;3 poss\u00edveis da matriz A, e como nenhum deles \u00e9 diferente de 0, <strong>a matriz n\u00e3o \u00e9 de posto 3<\/strong> . Portanto, no m\u00e1ximo ser\u00e1 o rank 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-157fd11377c30ccf66e64960e295866b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A) < 3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Veremos agora se a matriz \u00e9 de posto 2. Para isso, devemos encontrar uma submatriz quadrada de ordem 2 cujo determinante seja diferente de 0. Tentaremos a submatriz 2\u00d72 no 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-b0ae4ab76e4e45bbb1aecd49af2523a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; \\cellcolor[HTML]{ABEBC6}3 &amp; 4  &amp; -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp; &amp; \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp; \\cellcolor[HTML]{ABEBC6}2 &amp;  1 &amp; -1 &amp;  &amp; &amp; \\\\[-2ex] 3 &amp; -1 &amp;  7 &amp; 2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"75\" width=\"411\" 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-3320ea7301733c03681caf31e7539b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 3 \\\\[1.1ex] 0 &amp; 2  \\end{vmatrix} = 2-0 = 2 \\bm{ \\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"167\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Encontramos um determinante de ordem 2 diferente de 0 dentro da matriz. Consequentemente, <strong>a matriz \u00e9 de posto 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Problemas de escopo de matriz resolvidos<\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Determine a classifica\u00e7\u00e3o da seguinte matriz 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-ca5f88e86382a14720247e910084095c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; 1 \\\\[1.1ex] 5 &amp; 6  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" 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\"> Primeiro calculamos o determinante de toda a matriz:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88be02e3f0e84b30178b811354994424_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}=\\begin{vmatrix} 3 &amp; 1 \\\\[1.1ex] 5 &amp; 6 \\end{vmatrix} = 18-5 = 13 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"233\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Encontramos um determinante de ordem 2 diferente de 0. Portanto, <strong>a matriz \u00e9 de posto 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/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 a extens\u00e3o da seguinte matriz de dimens\u00e3o 2 \u00d7 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-19dde855da87ad73bdec3135fca04e78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 3 \\\\[1.1ex] 4 &amp; 6  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" 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\"> Primeiro, resolvemos o determinante de toda a matriz:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb383f77013e752e0f22ad582dbd3c80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}=\\begin{vmatrix} 2 &amp; 3 \\\\[1.1ex] 4 &amp; 6 \\end{vmatrix} = 12-12 \\bm{=0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"201\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O \u00fanico determinante 2\u00d72 poss\u00edvel d\u00e1 0, ent\u00e3o a matriz n\u00e3o \u00e9 de posto 2.<\/p>\n<p class=\"has-text-align-left\"> Mas dentro da matriz existem determinantes 1&#215;1 diferentes de 0, por exemplo:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9bfea6551282e7213ca85662eb657b6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2  \\end{vmatrix} = 2 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>A matriz \u00e9, portanto, de posto 1.<\/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-e9b93965a2d6e8834b62367fbe854e02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/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> Qual \u00e9 a extens\u00e3o da seguinte matriz quadrada 3&#215;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-fbe69cc53a58fd72117fa4aaa7a0ec38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; -3 &amp; 2 \\\\[1.1ex] 2 &amp; 1 &amp; 4 \\\\[1.1ex] 1 &amp; 4 &amp; 2 \\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\"> Primeiro, o determinante de toda a matriz \u00e9 calculado usando a regra de Sarrus:<\/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-a1bda19a46e006dfc43ade0e92f189e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix} 1 &amp; -3 &amp; 2 \\\\[1.1ex] 2 &amp; 1 &amp; 4 \\\\[1.1ex] 1 &amp; 4 &amp; 2 \\end{vmatrix} = 2-12+16-2-16+12 \\bm{=0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"380\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O \u00fanico determinante 3\u00d73 poss\u00edvel d\u00e1 0, ent\u00e3o a matriz n\u00e3o \u00e9 de posto 3.<\/p>\n<p class=\"has-text-align-left\"> Mas dentro da matriz existem determinantes de ordem 2 diferentes de 0, por exemplo:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9a1e82c35249f351ba9513437da95c65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 \\\\[1.1ex] 2 &amp; 1  \\end{vmatrix} = 1 +6 = 7 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"181\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, <strong>a matriz \u00e9 de posto 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/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> Calcule a classifica\u00e7\u00e3o da seguinte matriz 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-7d952325e084adb3fa3b97c7fc10c1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; -1 &amp; 1 \\\\[1.1ex] 4 &amp; -2 &amp; 3 \\\\[1.1ex] 2 &amp; 5 &amp; 2 \\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\"> Primeiro, o determinante de toda a matriz \u00e9 resolvido pela regra de Sarrus:<\/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-819e9bea5c6d6d536a4dafba325ae45e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix} 3 &amp; -1 &amp; 1 \\\\[1.1ex] 4 &amp; -2 &amp; 3 \\\\[1.1ex] 2 &amp; 5 &amp; 2 \\end{vmatrix} = -12-6+20+4-45+8 =  -31\\bm{ \\neq0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"440\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O determinante de toda a matriz \u00e9 avaliado como algo diferente de 0. Portanto, a matriz tem classifica\u00e7\u00e3o m\u00e1xima, ou seja, <strong>classifica\u00e7\u00e3o 3.<\/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-8548c1b53b3e4fbb5509589cb60f87b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=3}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/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> Qual \u00e9 a classifica\u00e7\u00e3o da seguinte matriz 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-d90091dd51727e806e6788a9594735ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 5 &amp; -1 \\\\[1.1ex] 3 &amp; -2 &amp; -4 \\\\[1.1ex] 5 &amp; 3 &amp; -5 \\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\"> Primeiro, o determinante de toda a matriz \u00e9 calculado usando a regra de Sarrus:<\/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-e4eee911dcf234c3fa63177e533901af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix}2 &amp; 5 &amp; -1 \\\\[1.1ex] 3 &amp; -2 &amp; -4 \\\\[1.1ex] 5 &amp; 3 &amp; -5 \\end{vmatrix} =20-100-9-10+24+75 \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"411\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O \u00fanico determinante 3\u00d73 poss\u00edvel d\u00e1 0, ent\u00e3o a matriz n\u00e3o \u00e9 de posto 3.<\/p>\n<p class=\"has-text-align-left\"> Mas dentro da matriz existem determinantes 2 \u00d7 2 diferentes de 0, como:<\/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-1397b7f935df1c8cce082c3f2f1418d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; 5 \\\\[1.1ex] 3 &amp; -2  \\end{vmatrix} = -4-15 = -19\\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"226\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>A matriz \u00e9, portanto, de posto 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 6<\/h3>\n<p> Encontre a extens\u00e3o da seguinte matriz 3&#215;4: <\/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-46ff20ee9ee9e4fac3e8858c55961f8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; 2 &amp; -4 &amp; 1 \\\\[1.1ex] 2 &amp; -2 &amp; -3 &amp; 5 \\\\[1.1ex] 5 &amp; 0 &amp; -7 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"175\" 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 n\u00e3o pode ser de posto 4, porque n\u00e3o podemos fazer determinantes 4\u00d74. Ent\u00e3o, vamos ver se \u00e9 de classifica\u00e7\u00e3o 3 calculando determinantes 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-2897851f49a9556fc03aded5f1495297_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix}3 &amp; 2 &amp; -4  \\\\[1.1ex] 2 &amp; -2 &amp; -3  \\\\[1.1ex] 5 &amp; 0 &amp; -7 \\end{vmatrix} =42-30+0-40-0+28 \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"393\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O determinante das 3 primeiras colunas d\u00e1 0. No entanto, o determinante das 3 \u00faltimas colunas d\u00e1 algo diferente de 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-c0821770a710807269d81fb1f8dd21a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; -4 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 5 \\\\[1.1ex] 0 &amp; -7 &amp; 3  \\end{vmatrix} = -18+0+14-0+70-24 = 42 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"400\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, como dentro existe uma submatriz de ordem 3 cujo determinante \u00e9 diferente de 0, <strong>a matriz \u00e9 de posto 3<\/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-8548c1b53b3e4fbb5509589cb60f87b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=3}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 7<\/h3>\n<p> Calcule o intervalo da seguinte matriz 4&#215;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-83e7cebc0d95d73f653cf54bd316c4f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 3 &amp; 4 &amp; -5  \\\\[1.1ex] 5 &amp; -2 &amp; -9  \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"164\" 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 n\u00e3o pode ser de posto 4, pois n\u00e3o podemos resolver nenhum determinante 4\u00d74. Ent\u00e3o vamos ver se \u00e9 de rank 3 fazendo todos os determinantes 3&#215;3 poss\u00edveis: <\/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-1eec1befc1515b4405529ede01c55618_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 3 &amp; 4 &amp; -5 \\\\[1.1ex] 5 &amp; -2 &amp; -9\\end{vmatrix} \\bm{= 0} \\qquad \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 5 &amp; -2 &amp; -9 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"308\" 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-fa1de344c0bb747c9861afd4de5fa7c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 3 &amp; 4 &amp; -5 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0} \\qquad \\begin{vmatrix} 3 &amp; 4 &amp; -5 \\\\[1.1ex] 5 &amp; -2 &amp; -9 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"322\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como todos os determinantes 3\u00d73 poss\u00edveis d\u00e3o 0, a matriz tamb\u00e9m n\u00e3o \u00e9 de posto 3. Tentamos agora os determinantes 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-981d085760dd1b1dd46aab17f1d7ba78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 \\\\[1.1ex] 3 &amp; 4  \\end{vmatrix} =13 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"127\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como dentro da matriz A existe uma submatriz de ordem 2 cujo determinante \u00e9 diferente de 0, <strong>a matriz \u00e9 de posto 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 8<\/h3>\n<p> Encontre o contradom\u00ednio da seguinte matriz 4 \u00d7 4: <\/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-bd5abef80b8d6ae74d4d60a0cf11e3ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 0 &amp; 1 &amp; -1  \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; -1  \\\\[1.1ex] 4 &amp; -2 &amp; -1 &amp; 3  \\\\[1.1ex] -1 &amp; 3 &amp; 2 &amp;  -4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" 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\"> Devemos resolver o determinante de toda a matriz para ver se ela \u00e9 de posto 4.<\/p>\n<p class=\"has-text-align-left\"> E para resolver o determinante 4&#215;4, voc\u00ea deve primeiro fazer opera\u00e7\u00f5es com as linhas para transformar todos os elementos de uma coluna, exceto um, em 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-27642038b0dc0358b382aaeab5c55263_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 2 &amp; 0 &amp; 1 &amp; -1 \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; -1 \\\\[1.1ex] 4 &amp; -2 &amp; -1 &amp; 3 \\\\[1.1ex] -1 &amp; 3 &amp; 2 &amp; -4 \\end{vmatrix} \\begin{matrix} \\\\[1.1ex]  \\\\[1.1ex]\\xrightarrow{f_3 + 2f_2} \\\\[1.1ex] \\xrightarrow{f_4 - 3f_2} \\end{matrix} \\begin{vmatrix} 2 &amp; 0 &amp; 1 &amp; -1 \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; -1 \\\\[1.1ex] 10 &amp; 0 &amp; 1 &amp; 1 \\\\[1.1ex] -10 &amp; 0 &amp; -1 &amp; -1 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"111\" width=\"360\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Calculamos agora o determinante por deputados:<\/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-239ee8aebdd8161e1e86d3d093ade490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 2 &amp; 0 &amp; 1 &amp; -1 \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; -1 \\\\[1.1ex] 10 &amp; 0 &amp; 1 &amp; 1 \\\\[1.1ex] -10 &amp; 0 &amp; -1 &amp; -1 \\end{vmatrix} \\displaystyle = 0\\bm{\\cdot} \\text{Adj(0)} +1\\bm{\\cdot} \\text{Adj(1)} +0\\bm{\\cdot} \\text{Adj(0)} + 0\\bm{\\cdot} \\text{Adj(0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"492\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Simplificamos os termos: <\/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-cd7140ff98995310b9c70e27c89dba05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"=\\cancel{0\\bm{\\cdot} \\text{Adj(0)}}+1\\bm{\\cdot} \\text{Adj(1)} +\\cancel{0\\bm{\\cdot} \\text{Adj(0)}} + \\cancel{0\\bm{\\cdot} \\text{Adj(0)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"343\" style=\"vertical-align: -5px;\"><\/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-029594698d2ffb9e165ed06c51bd495e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"= \\text{Adj(1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Calculamos o adjunto de 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-b30414c1569334502b1f17ee5380bd4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = (-1)^{2+2} \\begin{vmatrix}2 &amp;  1 &amp; -1 \\\\[1.1ex] 10 &amp; 1 &amp; 1 \\\\[1.1ex] -10 &amp; -1 &amp; -1\\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"202\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, por fim, calculamos o determinante 3\u00d73 com a regra de Sarrus e a calculadora: <\/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-15b1857a0769672f75e0ba922e34413a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = (-1)^{4} \\cdot \\bigl[-2-10+10-10+2+10 \\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"298\" style=\"vertical-align: -7px;\"><\/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-f1b44afd86030388f2b3eb74f2117708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = 1 \\cdot \\bigl[0 \\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -7px;\"><\/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-e0f707524d15b7f3351b2e331ca447cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O determinante 4&#215;4 de toda a matriz d\u00e1 0, ent\u00e3o a matriz A n\u00e3o ter\u00e1 classifica\u00e7\u00e3o 4. Ent\u00e3o agora vamos ver se ela tem um determinante 3&#215;3 diferente de 0 dentro:<\/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-8a8dabdc8197de8102d9e0c50db837a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; 0 &amp; 1  \\\\[1.1ex] 3 &amp; 1 &amp; 1  \\\\[1.1ex] 4 &amp; -2 &amp; -1  \\end{vmatrix} = -2+0-6-4+4-0=8 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"355\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>A matriz A \u00e9, portanto, de classifica\u00e7\u00e3o 3:<\/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-8548c1b53b3e4fbb5509589cb60f87b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=3}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"estil_titol_H2 wp-block-heading\"> Propriedades do intervalo de matrizes<\/h2>\n<ul>\n<li> O intervalo n\u00e3o \u00e9 modificado se excluirmos uma linha preenchida com zeros, seja uma coluna ou uma linha preenchida com 0.<\/li>\n<\/ul>\n<ul>\n<li> O contradom\u00ednio de uma matriz n\u00e3o muda se alterarmos a ordem de duas linhas paralelas, sejam elas linhas ou colunas.<\/li>\n<\/ul>\n<ul>\n<li> A classifica\u00e7\u00e3o de uma matriz \u00e9 igual \u00e0 de sua transposta.<\/li>\n<\/ul>\n<ul>\n<li> Se voc\u00ea multiplicar uma linha ou coluna por um n\u00famero diferente de 0, a classifica\u00e7\u00e3o da matriz n\u00e3o muda.<\/li>\n<\/ul>\n<ul>\n<li> O intervalo de uma tonalidade n\u00e3o muda quando eliminamos uma linha (linha ou coluna) que \u00e9 uma combina\u00e7\u00e3o linear de outras linhas paralelas a ela.<\/li>\n<\/ul>\n<ul>\n<li> O contradom\u00ednio de uma matriz n\u00e3o muda se adicionarmos outras linhas paralelas a qualquer uma das linhas (linhas ou colunas) multiplicadas por qualquer n\u00famero. \u00c9 por isso que a classifica\u00e7\u00e3o de uma matriz tamb\u00e9m pode ser calculada pelo m\u00e9todo gaussiano. <\/li>\n<\/ul>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 e como calcular a imagem de uma matriz por determinantes. Al\u00e9m disso, voc\u00ea encontrar\u00e1 exemplos e exerc\u00edcios resolvidos para aprender como encontrar facilmente a extens\u00e3o de uma matriz. Al\u00e9m disso, voc\u00ea tamb\u00e9m ver\u00e1 as propriedades de intervalo de uma matriz. Qual \u00e9 a classifica\u00e7\u00e3o de uma matriz? A &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/classificacao-de-uma-matriz\/\"> <span class=\"screen-reader-text\">Calcule a classifica\u00e7\u00e3o de uma matriz por determinantes<\/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":[12],"tags":[],"class_list":["post-293","post","type-post","status-publish","format-standard","hentry","category-determinante-de-uma-matriz"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Calcular a classifica\u00e7\u00e3o de uma matriz por determinantes - Matoridade<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/classificacao-de-uma-matriz\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Calcular a classifica\u00e7\u00e3o de uma matriz por determinantes - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea ver\u00e1 o que \u00e9 e como calcular a imagem de uma matriz por determinantes. Al\u00e9m disso, voc\u00ea encontrar\u00e1 exemplos e exerc\u00edcios resolvidos para aprender como encontrar facilmente a extens\u00e3o de uma matriz. Al\u00e9m disso, voc\u00ea tamb\u00e9m ver\u00e1 as propriedades de intervalo de uma matriz. Qual \u00e9 a classifica\u00e7\u00e3o de uma matriz? 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