{"id":302,"date":"2023-07-06T14:52:33","date_gmt":"2023-07-06T14:52:33","guid":{"rendered":"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/"},"modified":"2023-07-06T14:52:33","modified_gmt":"2023-07-06T14:52:33","slug":"teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/","title":{"rendered":"Teorema de rouche-fr\u00e9benius"},"content":{"rendered":"<p>Nesta p\u00e1gina descobriremos o que \u00e9 o <strong>teorema de Rouch\u00e9 Frobenius<\/strong> e como calcular o posto de uma matriz com ele. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos passo a passo com o teorema de Rouch\u00e9-Frobenius.<\/p>\n<h2 class=\"wp-block-heading\"> Qual \u00e9 o teorema de Rouch\u00e9-Frobenius?<\/h2>\n<p> <strong>O teorema de Rouch\u00e9-Frobenius \u00e9 um m\u00e9todo para classificar sistemas de equa\u00e7\u00f5es lineares.<\/strong> Em outras palavras, o teorema de Rouch\u00e9-Frobenius \u00e9 usado para descobrir quantas solu\u00e7\u00f5es um sistema de equa\u00e7\u00f5es tem sem precisar resolv\u00ea-lo.<\/p>\n<p> Existem 3 tipos de sistemas de equa\u00e7\u00f5es:<\/p>\n<ul>\n<li> <strong>Compat\u00edvel com Sistema Determinado (SCD):<\/strong> O sistema possui uma solu\u00e7\u00e3o \u00fanica.<\/li>\n<li> <strong>Sistema compat\u00edvel indeterminado (ICS):<\/strong> o sistema possui infinitas solu\u00e7\u00f5es.<\/li>\n<li> <strong>Sistema Incompat\u00edvel (SI):<\/strong> O sistema n\u00e3o tem solu\u00e7\u00e3o.<\/li>\n<\/ul>\n<p> Al\u00e9m disso, o teorema de Rouch\u00e9-Frobenius tamb\u00e9m nos permitir\u00e1 mais tarde <a href=\"https:\/\/mathority.org\/pt\/exemplos-de-regras-e-exercicios-resolvidos-de-cramer\/\">resolver sistemas usando a regra de Cramer<\/a> .<\/p>\n<h2 class=\"wp-block-heading\"> Declara\u00e7\u00e3o do teorema de Rouch\u00e9-Frobenius<\/h2>\n<p> O teorema de Rouch\u00e9-Frobenius diz que<\/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-bd767a13412c19de65e75a6826caee08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{A}\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 a matriz formada pelos coeficientes das inc\u00f3gnitas de um sistema de equa\u00e7\u00f5es. e a barriga<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bf22ad0d457d763be692e97f3bcdf221_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{A'}\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<p> , ou <strong>matriz estendida<\/strong> , \u00e9 a matriz formada pelos coeficientes das inc\u00f3gnitas de um sistema de equa\u00e7\u00f5es e pelos termos independentes: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\" style=\"margin-bottom:20px\"> O <strong>teorema de Rouch\u00e9-Frobenius<\/strong> permite-nos saber com que tipo de sistema de equa\u00e7\u00f5es estamos lidando de acordo com o posto das matrizes A e A&#8217;:<\/p>\n<ul style=\"color:#E53935; font-weight: bold;\">\n<li style=\"margin-bottom:20px\"> <span style=\"color:#000000;font-weight: normal;\">Se classifica\u00e7\u00e3o(A) = classifica\u00e7\u00e3o(A&#8217;) = n\u00famero de inc\u00f3gnitas \u27f6 Sistema compat\u00edvel determinado (SCD)<\/span><\/li>\n<li style=\"margin-bottom:20px;\"> <span style=\"color:#000000;font-weight: normal;\">Se classifica\u00e7\u00e3o (A) = classifica\u00e7\u00e3o (A&#8217;) &lt; n\u00famero de inc\u00f3gnitas \u27f6 Sistema compat\u00edvel indeterminado (SCI)<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">se intervalo (A)\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be0e48d5500c7e73c450241ea2197789_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\neq}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"13\" style=\"vertical-align: -4px;\"><\/p>\n<p><\/span> faixa (A&#8217;) \u27f6 Sistema incompat\u00edvel (SI)<\/li>\n<\/ul>\n<\/div>\n<p> Assim que soubermos o que diz o teorema de Rouch\u00e9-Frobenius, veremos como resolver os exerc\u00edcios do teorema de Rouch\u00e9-Frobenius. Aqui est\u00e3o 3 exemplos: um exerc\u00edcio resolvido usando o teorema de cada tipo de sistema de equa\u00e7\u00f5es.<\/p>\n<h2 class=\"wp-block-heading\"> Exemplo de sistema compat\u00edvel determinado (SCD)<\/h2>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b6b2f93c6308c25e8df2fbb5da2af9a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 2x+y-3z=0 \\\\[1.5ex] x+2y-z= 1 \\\\[1.5ex] 4x-2y+z = 3\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"135\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A <strong>matriz A<\/strong> e a <strong>matriz estendida A&#8217;<\/strong> do sistema s\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-4597f5171b586bbcf0915d8512f7b89d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 2 &amp; 1 &amp; -3  \\\\[1.1ex] 1 &amp; 2 &amp; -1  \\\\[1.1ex] 4 &amp; -2 &amp; 1  \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 2 &amp; 1 &amp; -3 &amp; 0 \\\\[1.1ex] 1 &amp; 2 &amp; -1 &amp; 1  \\\\[1.1ex] 4 &amp; -2 &amp; 1 &amp; 3\\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"405\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Calculamos agora a classifica\u00e7\u00e3o da matriz A. Para isso, verificamos se o determinante de toda a matriz \u00e9 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-6c95b7158a2e6401cd16aeb708f128ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 2 &amp; 1 &amp; -3  \\\\[1.1ex] 1 &amp; 2 &amp; -1  \\\\[1.1ex] 4 &amp; -2 &amp; 1  \\end{vmatrix} = 25 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"219\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como a matriz tem um determinante 3\u00d73 diferente de 0, <strong>a matriz A tem 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-842ae3b68df41813d9e409968f3ae946_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> Uma vez conhecida a classifica\u00e7\u00e3o de A, calculamos a classifica\u00e7\u00e3o de A&#8217;, que ser\u00e1 pelo menos a classifica\u00e7\u00e3o 3 porque acabamos de ver que tem dentro de si um determinante de ordem 3 diferente de 0. al\u00e9m disso, n\u00e3o pode ser de classifica\u00e7\u00e3o 4, j\u00e1 que n\u00e3o podemos fazer nenhum determinante de ordem 4. Portanto, <strong>a matriz A&#8217; tamb\u00e9m \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-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Assim, como o posto da matriz A \u00e9 igual ao posto da matriz A&#8217; e ao n\u00famero de inc\u00f3gnitas do sistema (3), sabemos pelo teorema de Rouch\u00e9 Frobenius que se trata de um <strong>Sistema Determinado Compat\u00edvel<\/strong> (SCD) :<\/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-557185e16670c72d23eec5a3ea13b487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 3 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = n = 3  \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCD}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"436\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Exemplo de um sistema compat\u00edvel indeterminado (ICS)<\/h2>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2360b9a47257f73cf3f5dea63fb24098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} x-y+2z=1 \\\\[1.5ex] 3x+2y+z= 5 \\\\[1.5ex] 2x+3y-z = 4\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"135\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A <strong>matriz A<\/strong> e a <strong>matriz estendida A&#8217;<\/strong> do sistema s\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-b281235e2702433b447e2586ae3092c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 1 &amp; -1 &amp; 2  \\\\[1.1ex] 3 &amp; 2 &amp; 1  \\\\[1.1ex] 2 &amp; 3 &amp; -1  \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 1 &amp; -1 &amp; 2 &amp; 1 \\\\[1.1ex] 3 &amp; 2 &amp; 1 &amp; 5  \\\\[1.1ex] 2 &amp; 3 &amp; -1 &amp; 4 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"405\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Calculamos agora a classifica\u00e7\u00e3o da matriz A. Para isso, verificamos se o determinante de toda a matriz \u00e9 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-74cafc27ab41134696c3bf263132b98b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 1 &amp; -1 &amp; 2  \\\\[1.1ex] 3 &amp; 2 &amp; 1  \\\\[1.1ex] 2 &amp; 3 &amp; -1 \\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"178\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O determinante de toda a matriz A d\u00e1 0, portanto n\u00e3o \u00e9 de posto 3. Para ver se \u00e9 de posto 2, devemos encontrar uma submatriz em A cujo determinante seja diferente de 0. Por exemplo, aquele 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-22b2487f7664a70c116593120de2743b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; -1  \\\\[1.1ex] 3 &amp; 2 \\end{vmatrix} = 5 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"123\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como a matriz tem um determinante 2\u00d72 diferente de 0, <strong>a matriz A tem 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Uma vez conhecida a classifica\u00e7\u00e3o de A, calculamos a classifica\u00e7\u00e3o de A&#8217;. J\u00e1 sabemos que o determinante das 3 primeiras colunas d\u00e1 0, ent\u00e3o tentamos os outros determinantes 3\u00d73 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-17f264ad3859da88ffa6784be24e4143_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}1 &amp; -1 &amp;  1 \\\\[1.1ex] 3 &amp; 2 &amp; 5  \\\\[1.1ex] 2 &amp; 3 &amp; 4\\end{vmatrix} = 0 \\quad \\begin{vmatrix}1 &amp; 2 &amp; 1 \\\\[1.1ex] 3 &amp;  1 &amp; 5  \\\\[1.1ex] 2 &amp; -1 &amp; 4\\end{vmatrix} = 0 \\quad \\begin{vmatrix} -1 &amp; 2 &amp; 1 \\\\[1.1ex] 2 &amp; 1 &amp; 5  \\\\[1.1ex] 3 &amp; -1 &amp; 4\\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"404\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Todos os determinantes 3\u00d73 da matriz A&#8217; s\u00e3o 0, ent\u00e3o a matriz A&#8217; tamb\u00e9m n\u00e3o ter\u00e1 classifica\u00e7\u00e3o 3. Por\u00e9m, dentro dele possui 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-22b2487f7664a70c116593120de2743b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; -1  \\\\[1.1ex] 3 &amp; 2 \\end{vmatrix} = 5 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"123\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto <strong>, a matriz A&#8217; ser\u00e1 de classifica\u00e7\u00e3o 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-80398cfd2fff647f81c0d4160f3b2f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A extens\u00e3o da matriz A \u00e9 igual \u00e0 extens\u00e3o da matriz A&#8217; mas estes s\u00e3o menores que o n\u00famero de inc\u00f3gnitas do sistema (3). Portanto, segundo o teorema de Rouch\u00e9-Frobenius, trata-se de um <strong>sistema compat\u00edvel indeterminado<\/strong> (ICS):<\/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-96868a2569ea0ab5ca99d8dc606d3dc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=2 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = 2 \\ < \\ n =3  \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"475\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Exemplo de sistema incompat\u00edvel (IS)<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-30e1084dd637eb4371f6b2218af24136_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 2x+y-2z=3 \\\\[1.5ex] 3x-2y+z= 2 \\\\[1.5ex] x+4-5z = 3 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"135\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p>A <strong>matriz A<\/strong> e a <strong>matriz estendida A&#8217;<\/strong> do sistema s\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-b435d86f1466af5748d91e6c9bd813e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 2 &amp; 1 &amp; -2 \\\\[1.1ex] 3 &amp; -2 &amp; 1 \\\\[1.1ex] 1 &amp; 4 &amp; -5 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 2 &amp; 1 &amp; -2 &amp; 3 \\\\[1.1ex] 3 &amp; -2 &amp; 1 &amp; 2  \\\\[1.1ex] 1 &amp; 4 &amp; -5 &amp; 3 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"405\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Calculamos agora a classifica\u00e7\u00e3o da matriz A. Para isso, verificamos se o determinante de toda a matriz \u00e9 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-714538c91aa2620a6adb40581245f0e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 2 &amp; 1 &amp; -2 \\\\[1.1ex] 3 &amp; -2 &amp; 1 \\\\[1.1ex] 1 &amp; 4 &amp; -5 \\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"178\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O determinante de toda a matriz A d\u00e1 0, portanto n\u00e3o \u00e9 de posto 3. Para ver se \u00e9 de posto 2, devemos encontrar uma submatriz em A cujo determinante seja diferente de 0. Por exemplo, aquele 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-5a46decda8fd850d9c847922b0c896db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 2 &amp; 1  \\\\[1.1ex] 3 &amp; -2 \\end{vmatrix} = -7 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Como a matriz possui um determinante de ordem 2 diferente de 0, <strong>a matriz A \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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Uma vez conhecida a classifica\u00e7\u00e3o de A, calculamos a classifica\u00e7\u00e3o de A&#8217;. J\u00e1 sabemos que o determinante das 3 primeiras colunas d\u00e1 0, ent\u00e3o agora tentamos, por exemplo, com o determinante das 3 \u00faltimas colunas:<\/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-47aecdf801b92f21f2287fb96eaaa3f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; -2 &amp; 3 \\\\[1.1ex]  -2 &amp; 1 &amp; 2  \\\\[1.1ex]  4 &amp; -5 &amp; 3 \\end{vmatrix} = 3 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"161\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Por outro lado, a matriz A&#8217; cont\u00e9m um determinante cujo resultado \u00e9 diferente de 0, portanto <strong>a matriz A&#8217; ter\u00e1 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-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, como o posto da matriz A \u00e9 menor que o posto da matriz A&#8217;, deduzimos do teorema de Rouch\u00e9-Frobenius que se trata de um <strong>Sistema Incompat\u00edvel<\/strong> (SI) <strong>:<\/strong> <\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3da0513f318d25473e93ba88c51fb42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = 2 \\ \\neq \\ rg(A') = 3 \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"426\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\">Problemas resolvidos do teorema de Rouch\u00e9-Frobenius <\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h3 class=\"estil_titol_H3 wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Determine o tipo do seguinte sistema de equa\u00e7\u00f5es com 3 inc\u00f3gnitas usando o teorema de Rouch\u00e9-Frobenius: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-du-theoreme-de-rouche-8211-frebenius-1.webp\" alt=\"Exerc\u00edcio resolvido do teorema de Rouche - frobenius\" class=\"wp-image-3984\" width=\"193\" height=\"122\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro fazemos a matriz A e a matriz estendida A&#8217; do sistema:<\/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-951ce5c1f0c606d4f060a1de58b60303_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 2 &amp; 1 &amp; -3 \\\\[1.1ex] 3 &amp; -1 &amp; -1 \\\\[1.1ex] -2 &amp; 4 &amp; 2 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 2 &amp; 1 &amp; -3 &amp; 0 \\\\[1.1ex] 3 &amp; -1 &amp; -1 &amp; 2 \\\\[1.1ex] -2 &amp; 4 &amp; 2 &amp; 8 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"432\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Devemos agora encontrar o posto da matriz A. Para isso, verificamos se o determinante da matriz \u00e9 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-15cddb69f7590648d1d6ae61d942471e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 2 &amp; 1 &amp; -3 \\\\[1.1ex] 3 &amp; -1 &amp; -1 \\\\[1.1ex] -2 &amp; 4 &amp; 2 \\end{vmatrix} = -4+2-36+6+8-6=-30 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"450\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A matriz tendo um determinante de terceira ordem diferente de 0, <strong>a matriz A tem 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-842ae3b68df41813d9e409968f3ae946_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 class=\"has-text-align-left\"> Uma vez conhecida a classifica\u00e7\u00e3o de A, calculamos a classifica\u00e7\u00e3o de A&#8217;. Este ser\u00e1 pelo menos de posto 3, porque acabamos de ver que tem dentro de um determinante de ordem 3 diferente de 0. Al\u00e9m disso, n\u00e3o pode ser de posto 4, pois n\u00e3o podemos deixar de fazer um determinante 4\u00d74. Portanto, <strong>a matriz A&#8217; tamb\u00e9m \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-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, gra\u00e7as ao teorema de Rouch\u00e9-Frobenius, sabemos que se trata de um <strong>sistema determinado compat\u00edvel<\/strong> (SCD), pois o contradom\u00ednio de A \u00e9 igual ao contradom\u00ednio de A&#8217; e ao n\u00famero de inc\u00f3gnitas. <\/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-31b495a48a75d7af1f23e38818bf4eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 3 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3 \\end{array}} \\\\ \\\\ \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = n = 3 \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCD}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"436\" 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> Classifique o seguinte sistema de equa\u00e7\u00f5es com 3 inc\u00f3gnitas usando o teorema de Rouch\u00e9-Frobenius: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-du-theoreme-de-rouche-8211-frebenius-2.webp\" alt=\"Exerc\u00edcio resolvido do teorema de Rouche-Frobenius\" class=\"wp-image-3987\" width=\"190\" height=\"121\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro constru\u00edmos a matriz A e a matriz estendida A&#8217; do sistema:<\/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-45e13aabe233ece927df7c9ba0bb3ec1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc}3 &amp; -1 &amp; 2  \\\\[1.1ex] 1 &amp; 2 &amp; -2  \\\\[1.1ex] 1 &amp; -5 &amp; 6 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 3 &amp; -1 &amp; 2 &amp; 1 \\\\[1.1ex] 1 &amp; 2 &amp; -2 &amp; 5 \\\\[1.1ex] 1 &amp; -5 &amp; 6 &amp; -9 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"419\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos calcular o intervalo da matriz A: <\/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-87bc95df0033834bba0398b8421faac5_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] 1 &amp; 2 &amp; -2 \\\\[1.1ex] 1 &amp; -5 &amp; 6 \\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-b9805283b75e2b89f67c7865a1263112_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; -1  \\\\[1.1ex] 1 &amp; 2 \\end{vmatrix} = 7 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"123\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, <strong>a matriz A tem classifica\u00e7\u00e3o 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Uma vez conhecida a classifica\u00e7\u00e3o de A, calculamos a classifica\u00e7\u00e3o de A&#8217;. J\u00e1 sabemos que o determinante das 3 primeiras colunas d\u00e1 0, ent\u00e3o tentamos os outros determinantes 3\u00d73 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-e6457fe3f03722b7f0d955191f318915_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}-1 &amp; 2 &amp; 1 \\\\[1.1ex] 2 &amp; -2 &amp; 5 \\\\[1.1ex] -5 &amp; 6 &amp; -9\\end{vmatrix} = 0 \\quad \\begin{vmatrix}3 &amp; 2 &amp; 1 \\\\[1.1ex] 1 &amp; -2 &amp; 5 \\\\[1.1ex] 1 &amp; 6 &amp; -9\\end{vmatrix} = 0 \\quad \\begin{vmatrix} 3 &amp; -1 &amp; 1 \\\\[1.1ex] 1 &amp; 2 &amp; 5 \\\\[1.1ex] 1 &amp; -5 &amp; -9\\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"446\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Todos os determinantes 3\u00d73 da matriz A&#8217; s\u00e3o 0, ent\u00e3o a matriz A&#8217; tamb\u00e9m n\u00e3o ter\u00e1 classifica\u00e7\u00e3o 3. Por\u00e9m, dentro dele possui muitos 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-eafa4747802fae3f0c36350357abbeb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} -1 &amp; 2  \\\\[1.1ex] 2 &amp; -2 \\end{vmatrix} = -2 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"150\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto <strong>, a matriz A&#8217; ser\u00e1 de classifica\u00e7\u00e3o 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-80398cfd2fff647f81c0d4160f3b2f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O posto da matriz A \u00e9 igual ao posto da matriz A&#8217; mas estes dois s\u00e3o menores que o n\u00famero de inc\u00f3gnitas do sistema (3). Portanto, pelo teorema de Rouch\u00e9-Frobenius sabemos que se trata de um <strong>sistema compat\u00edvel indeterminado<\/strong> (ICS): <\/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-96868a2569ea0ab5ca99d8dc606d3dc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=2 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = 2 \\ < \\ n =3  \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"475\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Determine que tipo de sistema \u00e9 o seguinte sistema de equa\u00e7\u00f5es usando o teorema de Rouch\u00e9-Frobenius: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-du-theoreme-de-rouche-8211-frebenius-3.webp\" alt=\"exerc\u00edcio resolvido passo a passo do teorema de rouche - frobenius\" class=\"wp-image-3990\" width=\"188\" height=\"122\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro fazemos a matriz A e a matriz estendida A&#8217; do sistema:<\/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-1820d31e4fd5c79804c9b6fa15abb469_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 1 &amp; 4 &amp; -2 \\\\[1.1ex] 3 &amp; -1 &amp; 3  \\\\[1.1ex] 5 &amp; 7 &amp; -1 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 1 &amp; 4 &amp; -2 &amp; 3 \\\\[1.1ex] 3 &amp; -1 &amp; 3 &amp; -2 \\\\[1.1ex] 5 &amp; 7 &amp; -1 &amp; 0 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"419\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos calcular o intervalo da matriz A: <\/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-4f998260ee4c96673085ea6fd4ca87ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 1 &amp; 4 &amp; -2 \\\\[1.1ex] 3 &amp; -1 &amp; 3 \\\\[1.1ex] 5 &amp; 7 &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-159a1c58fdcd972b4b08e4795950e064_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 4  \\\\[1.1ex] 3 &amp; -1 \\end{vmatrix} = -13 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, <strong>a matriz A tem classifica\u00e7\u00e3o 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Uma vez conhecida a classifica\u00e7\u00e3o de A, calculamos a classifica\u00e7\u00e3o de A&#8217;. J\u00e1 sabemos que o determinante das 3 primeiras colunas d\u00e1 0, mas n\u00e3o o determinante das 3 \u00faltimas colunas:<\/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-c673a5bbbd41933208169fa3e08b7c62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 4 &amp; -2 &amp; 3 \\\\[1.1ex]-1 &amp; 3 &amp; -2 \\\\[1.1ex] 7 &amp; -1 &amp; 0 \\end{vmatrix} = -40 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"198\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, <strong>a matriz A&#8217; tem 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-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O posto da matriz A \u00e9 menor que o posto da matriz A&#8217;, podemos portanto deduzir do teorema de Rouch\u00e9-Frobenius que se trata de um <strong>Sistema Incompat\u00edvel<\/strong> (SI) <strong>:<\/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-c3da0513f318d25473e93ba88c51fb42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = 2 \\ \\neq \\ rg(A') = 3 \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"426\" 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:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 4<\/h3>\n<p> Determine o tipo do seguinte sistema de equa\u00e7\u00f5es com 3 inc\u00f3gnitas usando o teorema de Rouch\u00e9-Frobenius: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exercice-resolu-du-theoreme-de-rouche-8211-frobenius-3-inconnues-3-equations.webp\" alt=\"Rouche - Teorema de Frobenius resolveu exerc\u00edcio com 3 inc\u00f3gnitas e 3 equa\u00e7\u00f5es\" class=\"wp-image-3991\" width=\"203\" height=\"122\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro fazemos a matriz A e a matriz estendida A&#8217; do sistema:<\/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-f8a0454c53a64f612c689ba1dae1196b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 5 &amp; -3 &amp; -2  \\\\[1.1ex] 1 &amp; 4 &amp; 1  \\\\[1.1ex]-3 &amp; 2 &amp; -2  \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 5 &amp; -3 &amp; -2 &amp; -2 \\\\[1.1ex] 1 &amp; 4 &amp; 1 &amp; 7 \\\\[1.1ex]-3 &amp; 2 &amp; -2 &amp; 3 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"446\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Devemos agora calcular a classifica\u00e7\u00e3o da matriz A. Para fazer isso, resolvemos o determinante da matriz 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-420f0d1ee000f39cbfbce88bf122f413_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 5 &amp; -3 &amp; -2 \\\\[1.1ex] 1 &amp; 4 &amp; 1 \\\\[1.1ex]-3 &amp; 2 &amp; -2 \\end{vmatrix} = -40+9-4-24-10-6=-75 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"467\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A matriz tendo um determinante de terceira ordem diferente de 0, <strong>a matriz A tem 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-842ae3b68df41813d9e409968f3ae946_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 class=\"has-text-align-left\"> Portanto, <strong>a matriz A&#8217; tamb\u00e9m \u00e9 de posto 3<\/strong> , pois \u00e9 sempre pelo menos de posto A e n\u00e3o pode ser de posto 4 porque n\u00e3o podemos resolver nenhum determinante 4\u00d74.<\/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-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, gra\u00e7as \u00e0 aplica\u00e7\u00e3o do teorema de Rouch\u00e9-Frobenius, sabemos que o sistema \u00e9 um <strong>Sistema Determinado Compat\u00edvel<\/strong> (SCD), pois o contradom\u00ednio de A \u00e9 igual ao contradom\u00ednio de A&#8217; e ao n\u00famero de inc\u00f3gnitas. <\/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-31b495a48a75d7af1f23e38818bf4eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 3 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3 \\end{array}} \\\\ \\\\ \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = n = 3 \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCD}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"436\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 5<\/h3>\n<p> Identifique que tipo de sistema \u00e9 o seguinte sistema de equa\u00e7\u00f5es usando o teorema de Rouch\u00e9-Frobenius: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-du-theoreme-de-rouche-8211-frebenius.webp\" alt=\"exemplo do teorema de Rouche - frobenius\" class=\"wp-image-3992\" width=\"205\" height=\"122\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro fazemos a matriz A e a matriz estendida A&#8217; do sistema:<\/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-3211e276b2b040969c38bc6c69eabd52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 4 &amp; -1 &amp; 3 \\\\[1.1ex] -1 &amp; 7 &amp; 3 \\\\[1.1ex] -5 &amp; 8 &amp; 0 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 4 &amp; -1 &amp; 3 &amp; 5 \\\\[1.1ex] -1 &amp; 7 &amp; 3 &amp; -3 \\\\[1.1ex] -5 &amp; 8 &amp; 0 &amp; 9 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"419\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos calcular o intervalo da matriz A: <\/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-21004095a3a8ef3edfc15bed5c7853a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 4 &amp; -1 &amp; 3 \\\\[1.1ex] -1 &amp; 7 &amp; 3 \\\\[1.1ex] -5 &amp; 8 &amp; 0\\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-a58059046b56cf1f8d82c6c8939e44ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 4 &amp; -1  \\\\[1.1ex]  -1 &amp; 7 \\end{vmatrix} = 27 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>A matriz A \u00e9, portanto, de classifica\u00e7\u00e3o 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Uma vez conhecida a classifica\u00e7\u00e3o de A, calculamos a classifica\u00e7\u00e3o de A&#8217;. O determinante das 3 primeiras colunas que j\u00e1 conhecemos d\u00e1 0, mas o determinante das 3 \u00faltimas colunas n\u00e3o d\u00e1:<\/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-992718d3b50aedf77c80c262fad5845f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} -1 &amp; 3 &amp; 5 \\\\[1.1ex]  7 &amp; 3 &amp; -3 \\\\[1.1ex] 8 &amp; 0 &amp; 9\\end{vmatrix} = -408 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"193\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, <strong>a matriz A&#8217; tem 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-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, por fim, aplicamos o dom\u00ednio ao teorema de Rouch\u00e9-Frobenius: o dom\u00ednio da matriz A \u00e9 menor que o dom\u00ednio da matriz A&#8217;, \u00e9 portanto um <strong>Sistema Incompat\u00edvel<\/strong> (SI) <strong>:<\/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-c3da0513f318d25473e93ba88c51fb42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = 2 \\ \\neq \\ rg(A') = 3 \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"426\" 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 6<\/h3>\n<p> Classifique o seguinte sistema de equa\u00e7\u00f5es de ordem 3 com o teorema de Rouch\u00e9-Frobenius: <\/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-d45e8bc425b08e403a98e01693201681_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 6x-2y+4z=1 \\\\[1.5ex] -2x+4y+3z= 7 \\\\[1.5ex] 8x-6y+z = -6\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"158\" 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 constru\u00edmos a matriz A e a matriz estendida A&#8217; do sistema:<\/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-8e779eca9135adc44e4a3a55f368560f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 6 &amp; -2 &amp; 4 \\\\[1.1ex] -2 &amp; 4 &amp; 3 \\\\[1.1ex] 8 &amp; -6 &amp; 1  \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 6 &amp; -2 &amp; 4 &amp; 1 \\\\[1.1ex] -2 &amp; 4 &amp; 3 &amp; 7 \\\\[1.1ex] 8 &amp; -6 &amp; 1 &amp; -6 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"419\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora vamos calcular o intervalo da matriz A: <\/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-c2f63f79858eae462547cf2f270fc780_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 6 &amp; -2 &amp; 4 \\\\[1.1ex] -2 &amp; 4 &amp; 3 \\\\[1.1ex] 8 &amp; -6 &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-e5fa293b94b8c6acfd998f1e154abf7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 6 &amp; -2  \\\\[1.1ex] -2 &amp; 4 \\end{vmatrix} = 20  \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, <strong>a matriz A tem classifica\u00e7\u00e3o 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Uma vez conhecida a classifica\u00e7\u00e3o de A, calculamos a classifica\u00e7\u00e3o de A&#8217;. J\u00e1 sabemos que o determinante das 3 primeiras colunas d\u00e1 0, ent\u00e3o tentamos os outros determinantes 3\u00d73 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-98958f866454a1bf9f1ac078562065cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} -2 &amp; 4 &amp; 1 \\\\[1.1ex]4 &amp; 3 &amp; 7 \\\\[1.1ex] -6 &amp; 1 &amp; -6\\end{vmatrix} = 0 \\quad \\begin{vmatrix}6 &amp; 4 &amp; 1 \\\\[1.1ex] -2 &amp; 3 &amp; 7 \\\\[1.1ex] 8 &amp;  1 &amp; -6\\end{vmatrix} = 0 \\quad \\begin{vmatrix} 6 &amp; -2 &amp; 1 \\\\[1.1ex] -2 &amp; 4 &amp; 7 \\\\[1.1ex] 8 &amp; -6 &amp; -6\\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"446\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Todos os determinantes 3\u00d73 da matriz A&#8217; s\u00e3o 0, ent\u00e3o a matriz A&#8217; tamb\u00e9m n\u00e3o ter\u00e1 classifica\u00e7\u00e3o 3. Por\u00e9m, dentro dele possui 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-58091f1a37a4ef81fdf56f01dd9531a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 6 &amp; -2 \\\\[1.1ex] -2 &amp; 4 \\end{vmatrix} = 20 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto <strong>, a matriz A&#8217; ser\u00e1 de classifica\u00e7\u00e3o 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-80398cfd2fff647f81c0d4160f3b2f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, aplicando o teorema de Rouch\u00e9-Frobenius, sabemos que se trata de um <strong>Sistema Compat\u00edvel Indeterminado<\/strong> (ICS), pois o contradom\u00ednio da matriz A \u00e9 igual ao contradom\u00ednio da matriz A&#8217; mas estes dois s\u00e3o menores que o n\u00famero de inc\u00f3gnitas no sistema (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-96868a2569ea0ab5ca99d8dc606d3dc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=2 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = 2 \\ < \\ n =3  \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"475\" 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 descobriremos o que \u00e9 o teorema de Rouch\u00e9 Frobenius e como calcular o posto de uma matriz com ele. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos passo a passo com o teorema de Rouch\u00e9-Frobenius. Qual \u00e9 o teorema de Rouch\u00e9-Frobenius? O teorema de Rouch\u00e9-Frobenius \u00e9 um m\u00e9todo para classificar sistemas de equa\u00e7\u00f5es lineares. &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Teorema de rouche-fr\u00e9benius<\/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":[24],"tags":[],"class_list":["post-302","post","type-post","status-publish","format-standard","hentry","category-sistemas-educacionais"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Teorema de Rouche-Fr\u00e9benius - 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\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Teorema de Rouche-Fr\u00e9benius - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina descobriremos o que \u00e9 o teorema de Rouch\u00e9 Frobenius e como calcular o posto de uma matriz com ele. Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos passo a passo com o teorema de Rouch\u00e9-Frobenius. Qual \u00e9 o teorema de Rouch\u00e9-Frobenius? O teorema de Rouch\u00e9-Frobenius \u00e9 um m\u00e9todo para classificar sistemas de equa\u00e7\u00f5es lineares. &hellip; Teorema de rouche-fr\u00e9benius Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T14:52:33+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd767a13412c19de65e75a6826caee08_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=\"9 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Teorema de rouche-fr\u00e9benius\",\"datePublished\":\"2023-07-06T14:52:33+00:00\",\"dateModified\":\"2023-07-06T14:52:33+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/\"},\"wordCount\":1725,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Sistemas educacionais\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/\",\"url\":\"https:\/\/mathority.org\/pt\/teorema-de-de-rouche-frobenius-com-exemplos-e-exercicios-resolvidos\/\",\"name\":\"Teorema de Rouche-Fr\u00e9benius - 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Voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos passo a passo com o teorema de Rouch\u00e9-Frobenius. Qual \u00e9 o teorema de Rouch\u00e9-Frobenius? 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