{"id":373,"date":"2026-10-19T08:15:20","date_gmt":"2026-10-19T06:15:20","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=373"},"modified":"2026-10-18T13:00:13","modified_gmt":"2026-10-18T11:00:13","slug":"alg-determinantes-factorizacion-lu-y-espacios-vectoriales-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=373","title":{"rendered":"ALG: Determinantes, factorizaci\u00f3n LU y espacios vectoriales con maxima"},"content":{"rendered":"<h2>Repaso de la inversa y pseudoinversa de una matriz<\/h2>\n<p>El procedimiento com\u00fan para el c\u00e1lculo de la inversa de una matriz (en caso de existir) puede plantearse como el algoritmo dado mediante transformaciones elementales:<\/p>\n<p>\\[[A\\, |\\, I_n] \\sim [I_n\\, |\\, A^{-1}].\\]<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Calcula mediante operaciones elementales la inversa de la matriz \\[\\begin{bmatrix}3 &#038; 0 &#038; -1 &#038; 1\\\\ -2 &#038; 3 &#038; 2 &#038; 2\\\\ 4 &#038; 2 &#038; -1 &#038; 1\\\\ -1 &#038; 2 &#038; -3 &#038; 0\\end{bmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3b1() {\n  var htmlShow3b1 = document.getElementById(\"html-show3b1\");\n  if (htmlShow3b1.style.display === \"none\") {\n    htmlShow3b1.style.display = \"block\";\n  } else {\n    htmlShow3b1.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3b1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3b1\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_inversa01.html\" width=\"650\" height=\"550\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\[A=\\begin{bmatrix}<br \/>\ni &#038; 1 &#038; -1 &#038; -i\\\\<br \/>\n0 &#038; i &#038; 1 &#038; 1 \\\\<br \/>\n0 &#038; 0 &#038; i &#038; -1 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; i<br \/>\n\\end{bmatrix}\\in\\mathcal{M}_4(\\mathbb{C}),\\]  \u00bfcu\u00e1nto suman los elementos de la cuarta columna de la inversa? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4() {\n  var htmlShow4 = document.getElementById(\"html-show4\");\n  if (htmlShow4.style.display === \"none\") {\n    htmlShow4.style.display = \"block\";\n  } else {\n    htmlShow4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGinv03.html\" width=\"650\" height=\"200\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\(A=\\begin{bmatrix} 1&#038;1&#038;0\\\\ 0&#038;1 &#038;1<br \/>\n\\end{bmatrix},\\) calcular su pseudoinversa por la derecha.\n<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3() {\n  var htmlShow3 = document.getElementById(\"html-show3\");\n  if (htmlShow3.style.display === \"none\") {\n    htmlShow3.style.display = \"block\";\n  } else {\n    htmlShow3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGpseudoinv03.html\" width=\"650\" height=\"200\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(A\\)=[[1,1],[1,0],[-1,0]], y \\(L\\) la pseudoinversa tal que \\(LA=I\\). \u00bfCu\u00e1nto suman los elementos de la segunda fila de \\(L\\)?\n<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3s() {\n  var htmlShow3s = document.getElementById(\"html-show3s\");\n  if (htmlShow3s.style.display === \"none\") {\n    htmlShow3s.style.display = \"block\";\n  } else {\n    htmlShow3s.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3s()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3s\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGpseudoinv01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<h3>Inversa en maxima<\/h3>\n<p>Hemos visto c\u00f3mo calcular la inversa mediante operaciones elementales; as\u00ed practicamos el uso de maxima y las matrices. Sin embargo, tenemos un comando que nos lo calcula directamente:<\/p>\n<ul>\n<li><strong>invert<\/strong>(\\(M\\)): dada la matriz \\(M\\) nos devuelve su inversa.<\/li>\n<li><strong>invert<\/strong>(\\(M\\)),<strong>detout<\/strong>: dada la matriz \\(M\\) nos devuelve su inversa con el determinante fuera.<\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\(A\\)=[[-1,1,3],[1,-2,0],[1,-2,1],[1,0,1]], estudiar si tiene pseudoinversa y determinarla en su caso<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv5() {\n  var htmlShow5 = document.getElementById(\"html-show5\");\n  if (htmlShow5.style.display === \"none\") {\n    htmlShow5.style.display = \"block\";\n  } else {\n    htmlShow5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5\" style=\"display: none;\">\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i2)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/>\n<\/span><span class=\"code_function\">rank<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o1} \\begin{bmatrix}\\mathop{-}1 &amp; 1 &amp; 3\\\\<br \/>\n1 &amp; \\mathop{-}2 &amp; 0\\\\<br \/>\n1 &amp; \\mathop{-}2 &amp; 1\\\\<br \/>\n1 &amp; 0 &amp; 1\\end{bmatrix}\\]\n<\/p>\n<p>\n\\[\\]\\[\\tag{%o2} 3\\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nComo el rango coincide con el n\u00famero de columnas, tiene pseudoinversa por la izquierda, que se obtendr\u00e1 mediante \\[L=(A^tA)^{-1}A^t.\\]\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i3)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">L<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">invert<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o3} \\begin{bmatrix}-\\frac{16}{59} &#038; -\\frac{5}{59} &#038; -\\frac{3}{59} &#038; \\frac{51}{59}\\\\<br \/>\n-\\frac{4}{59} &#038; -\\frac{16}{59} &#038; -\\frac{31}{118} &#038; \\frac{55}{118}\\\\<br \/>\n\\frac{15}{59} &#038; \\frac{1}{59} &#038; \\frac{13}{118} &#038; \\frac{15}{118}\\end{bmatrix}\\]\n<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\nEn efecto,\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n  <span class=\"prompt\"><br \/>\n(%i4)<br \/>\n  <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><br \/>\n<span class=\"code_variable\">L<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">;<\/span>  <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\n\\[\\]\\[\\tag{%o4} \\begin{bmatrix}1 &amp; 0 &amp; 0\\\\<br \/>\n0 &amp; 1 &amp; 0\\\\<br \/>\n0 &amp; 0 &amp; 1\\end{bmatrix}\\]\n<\/p>\n<\/div>\n<hr \/>\n<h2>Determinantes<\/h2>\n<p>Recordemos que para calcular un determinante hemos dado dos procesos: bien mediante triangulaci\u00f3n de la matriz, bien mediante la regla de Laplace. <\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Determina el valor del determinante de la matriz [[1,1,0,0], [-1,1,-1,0], [0,1,1,1], [0,0,1,1]], consiguiendo su matriz escalonada.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3b1q() {\n  var htmlShow3b1q = document.getElementById(\"html-show3b1q\");\n  if (htmlShow3b1q.style.display === \"none\") {\n    htmlShow3b1q.style.display = \"block\";\n  } else {\n    htmlShow3b1q.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3b1q()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3b1q\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Determinante mediante operaciones elementales con Maxima. Ej.14 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/wWmUknnjtbs?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<p>El siguiente comando nos proporciona directamente este resultado:<\/p>\n<ul>\n<li><strong>determinant<\/strong>(\\(M\\)): dada la matriz \\(M\\) nos devuelve su determinante.<\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Cu\u00e1l es el mayor valor de los menores principales de la matriz  \\[\\begin{bmatrix}<br \/>\n1 &#038; 4 &#038; -1 &#038; 1 &#038; 2 &#038; 1\\\\<br \/>\n8 &#038; 9 &#038; -2 &#038; 2 &#038;  -3&#038; 0\\\\<br \/>\n0 &#038; -3 &#038; 8 &#038; 3 &#038; 4 &#038; -3\\\\<br \/>\n4 &#038; -2 &#038; 0 &#038; -7 &#038; 1 &#038; 4\\\\<br \/>\n1 &#038; 0 &#038; -5 &#038; 3 &#038; 0&#038; 7\\\\<br \/>\n-3 &#038; 6 &#038; -4 &#038; 4 &#038; 6 &#038; 9<br \/>\n\\end{bmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv7() {\n  var htmlShow7 = document.getElementById(\"html-show7\");\n  if (htmlShow7.style.display === \"none\") {\n    htmlShow7.style.display = \"block\";\n  } else {\n    htmlShow7.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv7()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show7\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">9<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">9<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">mp<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">mp<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">mp<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">mp<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">mp<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">mp<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">append<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">mp<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\begin{bmatrix}1 &amp; 4\\\\8 &amp; 9\\end{bmatrix}, \\begin{bmatrix}1 &amp; 4 &amp; -1\\\\8 &amp; 9 &amp; -2\\\\0 &amp; -3 &amp; 8\\end{bmatrix}, \\begin{bmatrix}1 &amp; 4 &amp; -1 &amp; 1\\\\8 &amp; 9 &amp; -2 &amp; 2\\\\0 &amp; -3 &amp; 8 &amp; 3\\\\4 &amp; -2 &amp; 0 &amp; -7\\end{bmatrix}, \\begin{bmatrix}1 &amp; 4 &amp; -1 &amp; 1 &amp; 2\\\\8 &amp; 9 &amp; -2 &amp; 2 &amp; -3\\\\0 &amp; -3 &amp; 8 &amp; 3 &amp; 4\\\\4 &amp; -2 &amp; 0 &amp; -7 &amp; 1\\\\1 &amp; 0 &amp; -5 &amp; 3 &amp; 0\\end{bmatrix}\\right] \\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">makelist<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">mp<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ -23, -166, 986, 13768\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Cu\u00e1l es el valor de x para que el rango de la matriz sea 2  \\[\\begin{bmatrix}<br \/>\n5 &#038; -5 &#038; -6\\\\<br \/>\n-5 &#038; 3 &#038; -1 \\\\<br \/>\n0 &#038; x &#038;7<br \/>\n\\end{bmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv71() {\n  var htmlShow71 = document.getElementById(\"html-show71\");\n  if (htmlShow71.style.display === \"none\") {\n    htmlShow71.style.display = \"block\";\n  } else {\n    htmlShow71.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv71()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show71\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">ratsimp<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}5 &amp; -5 &amp; -6\\\\-5 &amp; 3 &amp; -1\\\\0 &amp; x &amp; 7\\end{bmatrix}\\]<\/p>\n<p>\\[\\operatorname{ }35 x-70\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ x=2\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Cu\u00e1l es el rango de la matriz seg\u00fan el valor del par\u00e1metro \\(\\alpha\\)  \\[\\begin{bmatrix}<br \/>\n\\alpha &#038; 1 &#038; 1 &#038; 2 \\\\<br \/>\n2 &#038; \\alpha &#038; \\alpha^2 &#038;1\\\\<br \/>\n2 &#038; 1 &#038; 1 &#038; 2<br \/>\n\\end{bmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv72() {\n  var htmlShow72 = document.getElementById(\"html-show72\");\n  if (htmlShow72.style.display === \"none\") {\n    htmlShow72.style.display = \"block\";\n  } else {\n    htmlShow72.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv72()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show72\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">^<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}x &amp; 1 &amp; 1 &amp; 2\\\\2 &amp; x &amp; {{x}^{2}} &amp; 1\\\\2 &amp; 1 &amp; 1 &amp; 2\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}2 &amp; 1 &amp; 1 &amp; 2\\\\2 &amp; x &amp; {{x}^{2}} &amp; 1\\\\x &amp; 1 &amp; 1 &amp; 2\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}2 &amp; 1 &amp; 1 &amp; 2\\\\0 &amp; x-1 &amp; {{x}^{2}}-1 &amp; -1\\\\0 &amp; 1-\\frac{x}{2} &amp; 1-\\frac{x}{2} &amp; 2-x\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Observar que si x=1<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">rank<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[3\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Como sabemos el rango para x=1, podemos hacer una divisi\u00f3n por x-1, suponiendo que x no es 1:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">ratsimp<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}2 &amp; 1 &amp; 1 &amp; 2\\\\0 &amp; x-1 &amp; {{x}^{2}}-1 &amp; -1\\\\0 &amp; 0 &amp; \\frac{{{x}^{2}}-2 x}{2} &amp; -\\frac{2 {{x}^{2}}-5 x+2}{2 x-2}\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">ratsimp<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[ -2 {{x}^{2}}+5 x-2\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ x=2\\operatorname{,}x=\\frac{1}{2}\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \u00bfQu\u00e9 valores de \\(x\\) hacen que la matriz no sea regular?  \\[\\begin{bmatrix}<br \/>\nx &#038; 1 &#038; -1\\\\<br \/>\n0 &#038; 2 &#038; x \\\\<br \/>\n4 &#038; 0 &#038; -x<br \/>\n\\end{bmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv171() {\n  var htmlShow171 = document.getElementById(\"html-show171\");\n  if (htmlShow171.style.display === \"none\") {\n    htmlShow171.style.display = \"block\";\n  } else {\n    htmlShow171.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv171()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show171\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-2 {{x}^{2}}+4 x+8\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[ \\left[ x=1-\\sqrt{5} \\ ,\\ x=\\sqrt{5}+1\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \u00bfQu\u00e9 valores de \\(x\\) hacen que la matriz no sea regular?  \\[\\begin{bmatrix}1&#038;3&#038;1&#038;3\\\\2&#038;3&#038;4&#038;5\\\\3&#038;2&#038;x&#038;2\\\\4&#038;x&#038;6&#038;5\\end{bmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv272() {\n  var htmlShow272 = document.getElementById(\"html-show272\");\n  if (htmlShow272.style.display === \"none\") {\n    htmlShow272.style.display = \"block\";\n  } else {\n    htmlShow272.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv272()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show272\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">ratsimp<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }{{x}^{2}}-8 x+15\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[ \\left[ x=3\\ ,\\ x=5\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<h2>Determinantes de una matriz por bloques<\/h2>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(A\\)=[ [1,6,7,0,0,0,0], [-3,-1,1,0,0,0,0], [0,4,-2,0,0,0,0],[1,1,0,5,-1,0,0], [0,3,7,-2,3,0,0], [-2,1,4,-2,-1,6,-5], [6,0,0,2,4,3,1]]. Calcular por bloques el valor de su determinante.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv272w() {\n  var htmlShow272w = document.getElementById(\"html-show272w\");\n  if (htmlShow272w.style.display === \"none\") {\n    htmlShow272w.style.display = \"block\";\n  } else {\n    htmlShow272w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv272w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show272w\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 6 &amp; 7 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\-3 &amp; -1 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 4 &amp; -2 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\1 &amp; 1 &amp; 0 &amp; 5 &amp; -1 &amp; 0 &amp; 0\\\\0 &amp; 3 &amp; 7 &amp; -2 &amp; 3 &amp; 0 &amp; 0\\\\-2 &amp; 1 &amp; 4 &amp; -2 &amp; -1 &amp; 6 &amp; -5\\\\6 &amp; 0 &amp; 0 &amp; 2 &amp; 4 &amp; 3 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Vemos claramente que podemos hacer la matriz por bloques, obteniendo una diagonal por bloques de manera que los bloques superiores son todos cero.<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">D1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">D2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">D3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 6 &amp; 7\\\\-3 &amp; -1 &amp; 1\\\\0 &amp; 4 &amp; -2\\end{bmatrix}\\]<\/p>\n<p>\\[\\begin{bmatrix}5 &amp; -1\\\\-2 &amp; 3\\end{bmatrix}\\]<\/p>\n<p>\\[\\begin{bmatrix}6 &amp; -5\\\\3 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Por tanto, el determinante ser\u00e1 el producto de los determinantes de los bloques de la diagonal.<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">D1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">D2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">D3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-33306\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Comprob\u00e9moslo<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-33306\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Factorizaci\u00f3n LU con maxima<\/h2>\n<p>Recordemos que las matrices L y U son \u00fanicas si la matriz no es singular (determinante distinto de cero), en contrario, pueden no ser \u00fanicas. Un ejemplo es cuando nos aparece un cero en la diagonal principal de la matriz U, en el proceso \\[[I\\, |\\, A] \\sim [L^*\\, |\\, U].\\] En tal caso debemos permutar las filas de la matriz \\(A\\) para que no ocurra. Pero si lo hacemos, debemos observar que ahora buscaremos una factorizaci\u00f3n de \\(PA\\) no de \\(A\\). Es decir, \\[PA=LU.\\]<\/p>\n<p>Es posible que proceso de factorizaci\u00f3n de la matriz \\(PA\\) nos lleve a la aparici\u00f3n de otro cero en la diagonal del escalonamiento de U; en ese caso, deberemos hacer otra permutaci\u00f3n. De este modo, la matriz \\(P\\) puede ser el resultado de un producto de matrices permutaci\u00f3n \\(P=P_1P_2\\cdots P_k\\). <\/p>\n<p>Recordemos que, si los menores principales de \\(A\\) son no nulos, el procedimiento nos lleva a una factorizaci\u00f3n. De este modo, podemos buscar la permutaci\u00f3n, o producto de permutaciones, adecuada de modo que los menores principales de \\(PA\\) sean no nulos y nos garanticen el \u00e9xito de la factorizaci\u00f3n.<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz [[1,2,3],[-3,-4,13],[2,1,-5]] determinar su factorizaci\u00f3n LU. <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4ae() {\n  var htmlShow4ae = document.getElementById(\"html-show4ae\");\n  if (htmlShow4ae.style.display === \"none\") {\n    htmlShow4ae.style.display = \"block\";\n  } else {\n    htmlShow4ae.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4ae()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4ae\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">13<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 1 &amp; 2 &amp; 3\\\\0 &amp; 1 &amp; 0 &amp; -3 &amp; -4 &amp; 13\\\\0 &amp; 0 &amp; 1 &amp; 2 &amp; 1 &amp; -5\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 1 &amp; 2 &amp; 3\\\\3 &amp; 1 &amp; 0 &amp; 0 &amp; 2 &amp; 22\\\\-2 &amp; 0 &amp; 1 &amp; 0 &amp; -3 &amp; -11\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 1 &amp; 2 &amp; 3\\\\3 &amp; 1 &amp; 0 &amp; 0 &amp; 2 &amp; 22\\\\\\frac{5}{2} &amp; \\frac{3}{2} &amp; 1 &amp; 0 &amp; 0 &amp; 22\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">U<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">L1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">L1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\3 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\\\frac{5}{2} &amp; \\frac{3}{2} &amp; 1 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i11)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">E<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0 &amp; -3 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 2 &amp; -\\frac{3}{2} &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i13)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">L<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">E<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">&quot;=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">L<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">\u00ab.\u00bb<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">U<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">&quot;=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">L<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">U<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\]\\[\\begin{bmatrix}1 &amp; 2 &amp; 3\\\\-3 &amp; -4 &amp; 13\\\\2 &amp; 1 &amp; -5\\end{bmatrix}=\\begin{bmatrix}1 &amp; 0 &amp; 0\\\\-3 &amp; 1 &amp; 0\\\\2 &amp; -\\frac{3}{2} &amp; 1\\end{bmatrix}.\\begin{bmatrix}1 &amp; 2 &amp; 3\\\\0 &amp; 2 &amp; 22\\\\0 &amp; 0 &amp; 22\\end{bmatrix}=\\begin{bmatrix}1 &amp; 2 &amp; 3\\\\-3 &amp; -4 &amp; 13\\\\2 &amp; 1 &amp; -5\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\[\\begin{bmatrix}<br \/>\n2  &#038; -1 &#038; 3 &#038; 1\\\\<br \/>\n1  &#038; -1 &#038; 2 &#038; 1\\\\<br \/>\n-1 &#038;  3 &#038; 2 &#038; 1\\\\<br \/>\n-1 &#038;  2 &#038; 3 &#038; 1<br \/>\n\\end{bmatrix},\\]  cu\u00e1nto suman todos los elementos de la matriz \\(L\\) de su factorizaci\u00f3n LU. <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2a3() {\n  var htmlShow2a3 = document.getElementById(\"html-show2a3\");\n  if (htmlShow2a3.style.display === \"none\") {\n    htmlShow2a3.style.display = \"block\";\n  } else {\n    htmlShow2a3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2a3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a3\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">Definici\u00f3n de la matriz original A<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i1)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Realicemos las operaciones elementales de fila para obtener la factorizaci\u00f3n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 2 &amp; -1 &amp; 3 &amp; 1\\\\0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 2 &amp; 1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; -1 &amp; 3 &amp; 2 &amp; 1\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 2 &amp; 3 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Eliminaci\u00f3n en Columna 5 (rowop resta c * F_j de F_i)<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 2 &amp; -1 &amp; 3 &amp; 1\\\\-\\frac{1}{2} &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; -\\frac{1}{2} &amp; \\frac{1}{2} &amp; \\frac{1}{2}\\\\\\frac{1}{2} &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; \\frac{5}{2} &amp; \\frac{7}{2} &amp; \\frac{3}{2}\\\\\\frac{1}{2} &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; \\frac{3}{2} &amp; \\frac{9}{2} &amp; \\frac{3}{2}\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Eliminaci\u00f3n en Columna 6<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">X2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 2 &amp; -1 &amp; 3 &amp; 1\\\\-\\frac{1}{2} &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; -\\frac{1}{2} &amp; \\frac{1}{2} &amp; \\frac{1}{2}\\\\-2 &amp; 5 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 6 &amp; 4\\\\-1 &amp; 3 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 6 &amp; 3\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Eliminaci\u00f3n en Columna 7<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X2<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">X2<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 2 &amp; -1 &amp; 3 &amp; 1\\\\-\\frac{1}{2} &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; -\\frac{1}{2} &amp; \\frac{1}{2} &amp; \\frac{1}{2}\\\\-2 &amp; 5 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 6 &amp; 4\\\\1 &amp; -2 &amp; -1 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; -1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Matriz U ser\u00e1<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">U<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}2 &amp; -1 &amp; 3 &amp; 1\\\\0 &amp; -\\frac{1}{2} &amp; \\frac{1}{2} &amp; \\frac{1}{2}\\\\0 &amp; 0 &amp; 6 &amp; 4\\\\0 &amp; 0 &amp; 0 &amp; -1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">La matriz L ser\u00e1 la inversa del resultado de las transformaciones en la identidad<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">L<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">invert<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">8<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0\\\\\\frac{1}{2} &amp; 1 &amp; 0 &amp; 0\\\\-\\frac{1}{2} &amp; -5 &amp; 1 &amp; 0\\\\-\\frac{1}{2} &amp; -3 &amp; 1 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Verifiqu\u00e9moslo<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i11) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">print<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">L<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">U<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_string\">&quot;=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">L<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">U<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\]\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0\\\\\\frac{1}{2} &amp; 1 &amp; 0 &amp; 0\\\\-\\frac{1}{2} &amp; -5 &amp; 1 &amp; 0\\\\-\\frac{1}{2} &amp; -3 &amp; 1 &amp; 1\\end{bmatrix} \\begin{bmatrix}2 &amp; -1 &amp; 3 &amp; 1\\\\0 &amp; -\\frac{1}{2} &amp; \\frac{1}{2} &amp; \\frac{1}{2}\\\\0 &amp; 0 &amp; 6 &amp; 4\\\\0 &amp; 0 &amp; 0 &amp; -1\\end{bmatrix}= \\begin{bmatrix}2 &amp; -1 &amp; 3 &amp; 1\\\\1 &amp; -1 &amp; 2 &amp; 1\\\\-1 &amp; 3 &amp; 2 &amp; 1\\\\-1 &amp; 2 &amp; 3 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Terminamos respondiendo a la pregunta<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i12) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">sum<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">sum<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">L<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">j<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }-\\frac{7}{2}\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Sistemas con LU<\/h2>\n<p>Uno de los usos est\u00e1 en la posibilidad de resolver sistemas de ecuaciones. Consideremos queremos resolver el sistema de ecuaciones \\[\\textbf{A}x=\\textbf{b},\\] donde \\(\\textbf{A}\\in\\mathcal{M}_{n\\times m}(\\mathbb{K})\\). Si conseguimos una factorizaci\u00f3n \\[\\textbf{A}=\\textbf{L}\\textbf{U},\\] donde \\(\\textbf{L}\\in\\mathcal{M}_{n\\times n}(\\mathbb{K})\\), y, \\(\\textbf{U}\\in\\mathcal{M}_{n\\times m}(\\mathbb{K})\\), resultar\u00e1<\/p>\n<p>\\[\\textbf{A}x=(\\textbf{L}\\textbf{U})x=\\textbf{L}(\\textbf{U}x)=\\textbf{b}.\\]<br \/>\nPara resolver el problema podemos afrontar la estrategia de resolver primero:<br \/>\n\\[\\textbf{L}y=\\textbf{b},\\] para despu\u00e9s<br \/>\n\\[\\textbf{U}x=\\textbf{y}.\\]<\/p>\n<p>Como ambas matrices \\(\\textbf{L}\\) y \\(\\textbf{U}\\) son triangulares, su soluci\u00f3n es f\u00e1cil mediante sustituci\u00f3n.<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Utilizar la factorizaci\u00f3n \\(\\textbf{LU}\\) anterior para resolver el sistema que permita resolver el sistema \\(Ax=\\textbf{b}\\) donde \\[A=\\begin{bmatrix}3 &#038; 2 &#038; -1\\\\ 2 &#038; 5 &#038; 6\\\\ -3 &#038; -2 &#038; 7 \\end{bmatrix},\\ \\textbf{b}=\\begin{bmatrix} 1\\\\ 3\\\\ -2\\end{bmatrix}\\]  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv435() {\n  var htmlShow435 = document.getElementById(\"html-show435\");\n  if (htmlShow435.style.display === \"none\") {\n    htmlShow435.style.display = \"block\";\n  } else {\n    htmlShow435.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv435()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show435\" style=\"display: none;\">\nEstar\u00e1 en breve. Hoy no, ma\u00f1ana.\n<\/div>\n<hr \/>\n<h2>Espacios vectoriales<\/h2>\n<h3>Complecio\u0301n de un conjunto l.i. a una base<\/h3>\n<p>A veces, necesitamos completar un conjunto l.i. de vectores para que formen una base de todo el espacio vectorial. Para conseguirlo podemos obrar de la siguiente forma. <\/p>\n<p>Consideremos \\(\\{\\textbf{v}_1,\\textbf{v}_2,\\ldots,\\textbf{v}_m\\}\\) un conjunto linealmente independiente de vectores de un espacio vectorial, \\(\\mathcal{V}\\), y \\(\\{\\textbf{e}_1,\\textbf{e}_2,\\ldots,\\textbf{e}_n\\}\\) una base de \\(\\mathcal{V}\\). Sea \\(A\\) la matriz cuyas filas son los vectores \\(\\textbf{v}_1,\\textbf{v}_2,\\ldots\\), \\(\\textbf{v}_m\\), y \\(B\\) la matriz de los vectores de la base, \\(\\textbf{e}_1,\\textbf{e}_2,\\ldots\\), \\(\\textbf{e}_n\\). Entonces realizamos operaciones elementales de forma que<br \/>\n\\[\\left[\\frac{A}{B}\\right] \\sim \\left[\\frac{E}{B^\\prime}\\right]\\] donde \\(E\\) es una matriz escalonada y \\({B^\\prime}\\) una matriz escalonada siguiente a \\(E\\), entonces los \\(n-m\\) escalones no nulos de \\({B^\\prime}\\) nos determinan los vectores que completan la base del espacio vectorial.<\/p>\n<blockquote><p><strong>Ejercicio:<\/strong> Sea \\(S=\\left\\{\\begin{bmatrix}a&#038;b\\\\c&#038;d\\end{bmatrix}\\in\\mathcal{M}_2(\\mathbb{R});a=b+c,\\ d=2c\\right\\}\\), completar su base para obtener una del espacio vectorial.\n<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2b1() {\n  var htmlShow2b1 = document.getElementById(\"html-show2b1\");\n  if (htmlShow2b1.style.display === \"none\") {\n    htmlShow2b1.style.display = \"block\";\n  } else {\n    htmlShow2b1.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2b1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2b1\" style=\"display: none;\">\nNecesitamos una base de \\(s\\) y para ello suficiente con observar que si \\(\\begin{bmatrix}a&#038;b\\\\c&#038;d\\end{bmatrix}\\in S\\) es:\\[ \\begin{bmatrix}a&#038;b \\\\c&#038;d\\end{bmatrix}= \\begin{bmatrix}b+c&#038;b\\\\ c&#038;2c\\end{bmatrix}= b\\begin{bmatrix}1&#038;1\\\\ 0&#038;0\\end{bmatrix}+c\\begin{bmatrix}1&#038;0\\\\ 1&#038;2\\end{bmatrix}\\]<br \/>\nLuego \\[\\left\\{\\begin{bmatrix}1&#038;1\\\\0&#038;0\\end{bmatrix},\\begin{bmatrix}1&#038;0\\\\1&#038;2\\end{bmatrix}\\right\\}\\] es una base del subespacio. Si consideramos la base can\u00f3nica de \\(\\mathcal{M}_2(\\mathbb{R})\\) podemos formar la matriz<br \/>\n\\[\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 1&#038;0&#038;1&#038;2\\\\ 1&#038;0&#038;0&#038;0\\\\ 0&#038;1&#038;0&#038;0\\\\ 0&#038;0&#038;1&#038;0\\\\ 0&#038;0&#038;0&#038;1 \\end{bmatrix}\\overset{f_2-f_1}{\\sim}\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 0&#038;-1&#038;1&#038;2\\\\ 1&#038;0&#038;0&#038;0\\\\ 0&#038;1&#038;0&#038;0\\\\ 0&#038;0&#038;1&#038;0\\\\ 0&#038;0&#038;0&#038;1 \\end{bmatrix}\\overset{f_3-f_1}{\\sim}\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 0&#038;-1&#038;1&#038;2\\\\ 0&#038;-1&#038;0&#038;0\\\\ 0&#038;1&#038;0&#038;0\\\\ 0&#038;0&#038;1&#038;0\\\\ 0&#038;0&#038;0&#038;1 \\end{bmatrix}\\]<br \/>\n\\[\\overset{f_3-f_2}{\\sim}\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 0&#038;-1&#038;1&#038;2\\\\ 0&#038;0&#038;-1&#038;-2\\\\ 0&#038;1&#038;0&#038;0\\\\ 0&#038;0&#038;1&#038;0\\\\ 0&#038;0&#038;0&#038;1 \\end{bmatrix}\\overset{f_4+f_2}{\\sim}\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 0&#038;-1&#038;1&#038;2\\\\ 0&#038;0&#038;-1&#038;-2\\\\ 0&#038;0&#038;1&#038;2\\\\ 0&#038;0&#038;1&#038;0\\\\ 0&#038;0&#038;0&#038;1 \\end{bmatrix}\\overset{f_4+f_3}{\\sim}\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 0&#038;-1&#038;1&#038;2\\\\ 0&#038;0&#038;-1&#038;-2\\\\ 0&#038;0&#038;0&#038;0\\\\ 0&#038;0&#038;1&#038;0\\\\ 0&#038;0&#038;0&#038;1 \\end{bmatrix}\\]<br \/>\n\\[\\overset{f_4\\leftrightarrow f_5}{\\sim}\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 0&#038;-1&#038;1&#038;2\\\\ 0&#038;0&#038;-1&#038;-2\\\\  0&#038;0&#038;1&#038;0\\\\ 0&#038;0&#038;0&#038;0\\\\0&#038;0&#038;0&#038;1 \\end{bmatrix}\\overset{f_5\\leftrightarrow f_6}{\\sim}\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 0&#038;-1&#038;1&#038;2\\\\ 0&#038;0&#038;-1&#038;-2\\\\  0&#038;0&#038;1&#038;0\\\\0&#038;0&#038;0&#038;1\\\\ 0&#038;0&#038;0&#038;0 \\end{bmatrix}\\overset{f_4+ f_3}{\\sim}\\begin{bmatrix}1&#038;1&#038;0&#038;0\\\\ 0&#038;-1&#038;1&#038;2\\\\ 0&#038;0&#038;-1&#038;-2\\\\  0&#038;0&#038;0&#038;-2\\\\0&#038;0&#038;0&#038;1\\\\ 0&#038;0&#038;0&#038;0 \\end{bmatrix}\\]<br \/>\nPor comodidad, podr\u00edamos permutar las filas 4 y 5 y tendr\u00edamos las cuatro primeras filas linealmente independientes. Luego \\[\\left\\{\\begin{bmatrix}1&#038;1\\\\0&#038;0\\end{bmatrix},\\begin{bmatrix}1&#038;0\\\\1&#038;2\\end{bmatrix},\\begin{bmatrix}0&#038;0\\\\-1&#038;-2\\end{bmatrix},\\begin{bmatrix}0&#038;0\\\\0&#038;1\\end{bmatrix}\\right\\}\\]\n<\/div>\n<hr\/>\n<blockquote><p><strong>Ejercicio:<\/strong> Sea \\(S=\\left\\{a+bX+cX^2+dX^3+eX^4\\in\\mathbb{R}_4[X];b=a-d,\\ c=e-2d\\right\\}\\), completar su base para obtener una del espacio vectorial \\(\\mathbb{R}_4[X]\\).\n<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2b13e() {\n  var htmlShow2b13e = document.getElementById(\"html-show2b13e\");\n  if (htmlShow2b13e.style.display === \"none\") {\n    htmlShow2b13e.style.display = \"block\";\n  } else {\n    htmlShow2b13e.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2b13e()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2b13e\" style=\"display: none;\">\nPrimero debemos sacar una base:<br \/>\n\\[S=\\left\\{a+bX+cX^2+dX^3+eX^4\\in\\mathbb{R}_4[X];b=a-d,\\ c=e-2d\\right\\}=\\] \\[=\\left\\{a+(a-d)X+(e-2d)X^2+dX^3+eX^4;a,d,e\\in\\mathbb{R}\\right\\}\\]<br \/>\nLuego \\[a+(a-d)X+(e-2d)X^2+dX^3+eX^4=a(1-X)+d(-X-2X^2+X^3)+e(X^2+X^4).\\] Es decir,<br \/>\n\\[S=\\mathbf{Gen}\\left\\{1-X,-X-2X^2+X^3,X^2+X^4\\right\\}\\]<br \/>\nPas\u00e9moslo a sus vectores de coordenadas y realicemos el proceso de completaci\u00f3n.<br \/>\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">w<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">w<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; -1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; -1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 1\\\\1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; -1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; -1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 1\\\\0 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; -1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; -1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 1\\\\0 &amp; 0 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">8<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; -1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; -1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 1\\\\0 &amp; 0 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i12) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[{ }\\begin{bmatrix}1 &amp; -1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; -1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 1\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 2\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; -1\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\end{bmatrix}\\]<\/p>\n<p>Aqu\u00ed podemos parar, pues los cinco primeros vectores son l.i.; es decir, una base de \\(\\mathbb{R}^4\\). Por tanto, para completar la base solo necesitamos los dos vectores que contin\u00faan a los que tenemos: \\[\\left\\{1-X,-X-2X^2+X^3,X^2+X^4, X^3+2X^4, -X^4\\right\\}\\]\n<\/p><\/div>\n<hr\/>\n<p>La complementaci\u00f3n que acabamos de realizar se fundamenta en el siguiente teorema:<\/p>\n<blockquote><p><strong>Teorema:<\/strong> Sea \\(S\\) un subespacio vectorial de un espacio vectorial \\(V\\), finitamente generado (e.v.f.g). Si \\(B_S\\) es una base de \\(S\\), existen \\(\\{u_1,\\ldots,u_r\\}\\subset V-B_S\\), de modo que \\(B_S\\cap \\{u_1,\\ldots,u_r\\}\\) es una base de \\(V\\).\n<\/p><\/blockquote>\n<p>As\u00ed podemos dar la siguiente definici\u00f3n:<\/p>\n<blockquote><p><strong>Definici\u00f3n:<\/strong> Sea \\(S\\) un subespacio vectorial de un espacio vectorial \\(V\\), e.v.f.g. Se llama subespacio vectorial <strong>suplementario<\/strong> de \\(S\\), con base \\(B_S\\), al subespacio generado por los vectores \\(\\{u_1,\\ldots,u_r\\}\\subset V-B_S\\), tales que \\(B_S\\cap \\{u_1,\\ldots,u_r\\}\\) es una base de \\(V\\).<\/p>\n<p>Observar que \\[S+\\mathbf{Gen}\\{u_1,\\ldots,u_r\\}=V\\]<\/p>\n<p>Se dice que la suma anterior es directa \\[S\\oplus\\mathbf{Gen}\\{u_1,\\ldots,u_r\\}=V,\\] cuando para cada \\(v\\in V\\) existen dos \u00fanicos \\(s\\in S\\) y \\(u\\in \\mathbf{Gen}\\{u_1,\\ldots,u_r\\}\\), tales que \\[s+u=v.\\]\n<\/p><\/blockquote>\n<blockquote><p><strong>Ejercicio:<\/strong> Sea \\(S=\\left\\{\\begin{bmatrix}a&#038;2b\\\\a-b&#038;a+b\\end{bmatrix};a,b\\in\\mathbb{R}\\right\\}\\subset\\mathcal{M}_2(\\mathbb{R})\\), obtener una base para su suplementario.\n<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1b2w() {\n  var htmlShow1b2w = document.getElementById(\"html-show1b2w\");\n  if (htmlShow1b2w.style.display === \"none\") {\n    htmlShow1b2w.style.display = \"block\";\n  } else {\n    htmlShow1b2w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1b2w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1b2w\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Compleci\u00f3n de una base de un subespacio vectorial Ej.1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/OK9mV29OrDA?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr\/>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\"><strong>Ejercicio:<\/strong> Dada la aplicaci\u00f3n lineal \\(f:\\mathbb{R}_2[X]\\to\\mathbb{R}_2[X]\\) dada por \\[\\forall p(X)\\in\\mathbb{R}_2[X];\\ f(p(X))=p(X)-\\frac{d}{dX}p(X),\\] y \\(M_f\\) su matriz asociada respecto de la base \\(B=\\{2+X-X^2,2X-1,1+3X^2\\}\\). \u00bfCu\u00e1l es el valor de \\(\\mathbf{tr}(M_fM_f^t)\\) ?<\/p>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>58<\/li>\n<li>64<\/li>\n<li>70<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<strong>C)<\/strong>.<br \/>\nComo \\[\\forall p(X)\\in\\mathbb{R}_2[X];\\ f(p(X))=p(X)-\\frac{d}{dX}p(X),\\]<br \/>\naplicamos cada uno de los polinomios de \\(B=\\{p=2+X-X^2,q=2X-1,r=1+3X^2\\}\\) tendremos<br \/>\n\\[f(p)=-{{X}^{2}}+3 X+1\\operatorname{,}f(q)=2 X-3\\operatorname{,}f(r)=3 {{X}^{2}}-6 X+1 \\]<br \/>\nAhora solo tenemos que colocar las coordenadas en columnas:<br \/>\n\\[M_f\\begin{bmatrix}1 &#038; -3 &#038; 1\\\\<br \/>\n3 &#038; 2 &#038; -6\\\\<br \/>\n-1 &#038; 0 &#038; 3\\end{bmatrix}\\]<br \/>\nResolvemos el ejercicio<br \/>\n\\[\\begin{bmatrix}1 &#038; -3 &#038; 1\\\\<br \/>\n3 &#038; 2 &#038; -6\\\\<br \/>\n-1 &#038; 0 &#038; 3\\end{bmatrix}\\begin{bmatrix}1 &#038; 3 &#038; -1\\\\<br \/>\n-3 &#038; 2 &#038; 0\\\\<br \/>\n1 &#038; -6 &#038; 3\\end{bmatrix}=\\begin{bmatrix}11 &#038; -9 &#038; 2\\\\<br \/>\n-9 &#038; 49 &#038; -21\\\\<br \/>\n2 &#038; -21 &#038; 10\\end{bmatrix}\\]<br \/>\nAs\u00ed la traza es 70.\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Repaso de la inversa y pseudoinversa de una matriz El procedimiento com\u00fan para el c\u00e1lculo de la inversa de una matriz (en caso de existir) puede plantearse como el algoritmo dado mediante&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[20],"class_list":["post-373","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-practicas-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/373","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=373"}],"version-history":[{"count":7,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/373\/revisions"}],"predecessor-version":[{"id":380,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/373\/revisions\/380"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=373"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=373"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=373"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}