{"id":171,"date":"2025-10-10T08:15:28","date_gmt":"2025-10-10T06:15:28","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=171"},"modified":"2025-10-09T19:14:09","modified_gmt":"2025-10-09T17:14:09","slug":"alg-inversa-de-una-matriz-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=171","title":{"rendered":"ALG: Determinantes y factorizaci\u00f3n LU 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 talque \\(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{pmatrix}5 &amp; -5 &amp; -6\\\\-5 &amp; 3 &amp; -1\\\\0 &amp; x &amp; 7\\end{pmatrix}\\]<\/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<h3>M\u00e1s comandos interesantes:<\/h3>\n<ul>\n<li><strong>adjoint<\/strong>(\\(M\\)): dada la matriz \\(M\\) nos devuelve su adjunta.<\/li>\n<li><strong>minor<\/strong>(\\(M,i,j\\)): dada la matriz \\(M\\) nos devuelve el menor (<em>i<\/em>,<em>j<\/em>), esto es, elimina la fila <em>i<\/em> y la columna <em>j<\/em> de la matriz.<\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\[A=\\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; 0 &#038; 0 &#038; \\cdots  &#038; 0 &#038; 0\\\\<br \/>\n2 &#038; 1 &#038; 0 &#038; 0 &#038; \\cdots  &#038; 0 &#038; 0\\\\<br \/>\n3 &#038; 2 &#038; 1 &#038; 0 &#038; \\cdots  &#038; 0 &#038; 0\\\\<br \/>\n4 &#038; 3 &#038; 2 &#038; 1 &#038; \\cdots &#038; 0 &#038; 0\\\\<br \/>\n\\vdots &#038; \\vdots  &#038; \\vdots  &#038;\\vdots   &#038; \\cdots  &#038; \\vdots  &#038; \\vdots \\\\<br \/>\nn-1 &#038; n-2 &#038; n-3 &#038; n-4 &#038; \\cdots  &#038; 1 &#038; 0\\\\<br \/>\nn &#038; n-1 &#038; n-2 &#038; n-3 &#038; \\cdots  &#038; 2 &#038; 1<br \/>\n\\end{bmatrix}\\in\\mathcal{M}_n(R),\\] y \\(adj(A)=[A_{ij}]^t\\) su matriz adjunta. \u00bfCu\u00e1ntos \\(A_{ij}=0\\) hay? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4w() {\n  var htmlShow4w = document.getElementById(\"html-show4w\");\n  if (htmlShow4w.style.display === \"none\") {\n    htmlShow4w.style.display = \"block\";\n  } else {\n    htmlShow4w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4w\" 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\">0<\/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_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">adjoint<\/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>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0\\\\-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\">(%i4)<\/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\">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_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\">0<\/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_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_function\">adjoint<\/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>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0\\\\-2 &amp; 1 &amp; 0\\\\1 &amp; -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\">(%i6)<\/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\">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_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\">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_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_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_operator\">[<\/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\">2<\/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 \/><\/span><span class=\"code_function\">adjoint<\/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>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0\\\\-2 &amp; 1 &amp; 0 &amp; 0\\\\1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 1 &amp; -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\">(%i8)<\/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\">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_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\">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_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_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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/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\">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_operator\">[<\/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_number\">2<\/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 \/><\/span><span class=\"code_function\">adjoint<\/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>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\-2 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\1 &amp; -2 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; -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\">(%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\">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\">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_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\">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_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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <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_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_operator\">[<\/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\">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_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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/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_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_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_number\">2<\/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 \/><\/span><span class=\"code_function\">adjoint<\/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>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\-2 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\1 &amp; -2 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; -2 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; -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\">(%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\">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\">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_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_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\">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_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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <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_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_operator\">[<\/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\">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_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\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <span class=\"code_operator\">[<\/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_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_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\">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_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> \u00a0\u00a0 <span class=\"code_operator\">[<\/span><span class=\"code_number\">7<\/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_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\">2<\/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 \/><\/span><span class=\"code_function\">adjoint<\/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>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\-2 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\1 &amp; -2 &amp; 1 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; -2 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; -2 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; -2 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; -2 &amp; 1\\end{bmatrix}\\]<\/p>\n<\/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\\) sea no nulos y nos garantice 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\">\u00ab=\u00bb<\/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\">\u00ab=\u00bb<\/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 \/>\n3 &#038; 2 &#038; -1\\\\<br \/>\n2 &#038; 4 &#038; 6\\\\<br \/>\n-1 &#038; 0 &#038; 3<br \/>\n\\end{bmatrix},\\]  cu\u00e1nto suman todos los elementos de la primera columna de la matriz \\(L\\) de su factorizaci\u00f3n LU. <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4a() {\n  var htmlShow4a = document.getElementById(\"html-show4a\");\n  if (htmlShow4a.style.display === \"none\") {\n    htmlShow4a.style.display = \"block\";\n  } else {\n    htmlShow4a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4a\" style=\"display: none;\">\nSoluci\u00f3n: \\(\\frac{4}{3}\\)\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\\[L=\\begin{bmatrix}1 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n\\frac{1}{2} &#038; 1 &#038; 0 &#038; 0\\\\<br \/>\n-\\frac{1}{2} &#038; -5 &#038; 1 &#038; 0\\\\<br \/>\n-\\frac{1}{2} &#038; -3 &#038; 1 &#038; 1\\end{bmatrix}\\]<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\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_function\">sum<\/span>(<span class=\"code_function\">sum<\/span>(<span class=\"code_variable\">L<\/span>[<span class=\"code_variable\">j<\/span>,<span class=\"code_variable\">i<\/span>],<span class=\"code_variable\">i<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">4<\/span>),<span class=\"code_variable\">j<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">4<\/span>)<span class=\"code_endofline\">;<\/span> <\/span><\/td>\n<\/tr>\n<\/table>\n<p> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(%o2) <\/mtext> <\/mtd> <mtd><mi>\u2212<\/mi><mfrac><mn>7<\/mn><mn>2<\/mn><\/mfrac> <\/mtd><\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Encontrar la factorizaci\u00f3n LU de la matriz \\[\\begin{bmatrix}  1 &#038;  3 &#038;-1 &#038; 2\\\\  2 &#038;  6 &#038;-1 &#038; 3\\\\  1 &#038;  3 &#038;-2 &#038; 4\\\\  3 &#038;  7 &#038;-2 &#038; 4 \\end{bmatrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv43w2() {\n  var htmlShow43w2 = document.getElementById(\"html-show43w2\");\n  if (htmlShow43w2.style.display === \"none\") {\n    htmlShow43w2.style.display = \"block\";\n  } else {\n    htmlShow43w2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv43w2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show43w2\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Factorizaci\u00f3n LU. Ej.9 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/DOgnbncdOD4?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<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\[\\begin{bmatrix}3 &#038; 2 &#038; -1\\\\ 2 &#038; 5 &#038; 6\\\\ -3 &#038; -2 &#038; 7 \\end{bmatrix},\\] determina su factorizaci\u00f3n LU.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv43w() {\n  var htmlShow43w = document.getElementById(\"html-show43w\");\n  if (htmlShow43w.style.display === \"none\") {\n    htmlShow43w.style.display = \"block\";\n  } else {\n    htmlShow43w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv43w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show43w\" style=\"display: none;\">\n<p> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable>  <mlabeledtr columnalign=\"center\"> <mtd><mtext>(A)<\/mtext> <\/mtd> <mtd><mrow>  <mo>(<\/mo>  <mrow> <mtable><mtr>  <mtd> <mn>3<\/mn>  <\/mtd>  <mtd> <mn>2<\/mn>  <\/mtd>  <mtd> <mrow><mi>\u2212<\/mi><mn>1<\/mn> <\/mrow>  <\/mtd><\/mtr><mtr>  <mtd> <mn>2<\/mn>  <\/mtd>  <mtd> <mn>5<\/mn>  <\/mtd>  <mtd> <mn>6<\/mn>  <\/mtd><\/mtr><mtr>  <mtd> <mrow><mi>\u2212<\/mi><mn>3<\/mn> <\/mrow>  <\/mtd>  <mtd> <mrow><mi>\u2212<\/mi><mn>2<\/mn> <\/mrow>  <\/mtd>  <mtd> <mn>7<\/mn>  <\/mtd><\/mtr> <\/mtable>  <\/mrow>  <mo>)<\/mo><\/mrow> <\/mtd>  <\/mlabeledtr>  <mlabeledtr columnalign=\"center\"> <mtd><mtext\/> <\/mtd> <mtd><mo>0 errores, 0 advertencias<\/mo> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable>  <mlabeledtr columnalign=\"center\"> <mtd><mtext>(%o3) <\/mtext> <\/mtd> <mtd><mo>[<\/mo><mrow>  <mo>(<\/mo>  <mrow> <mtable><mtr>  <mtd> <mn>1<\/mn>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd><\/mtr><mtr>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mn>1<\/mn>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd><\/mtr><mtr>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mn>1<\/mn>  <\/mtd><\/mtr> <\/mtable>  <\/mrow>  <mo>)<\/mo><\/mrow><mo>,<\/mo><mrow>  <mo>(<\/mo>  <mrow> <mtable><mtr>  <mtd> <mn>1<\/mn>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd><\/mtr><mtr>  <mtd> <mfrac><mn>2<\/mn><mn>3<\/mn> <\/mfrac>  <\/mtd>  <mtd> <mn>1<\/mn>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd><\/mtr><mtr>  <mtd> <mrow><mi>\u2212<\/mi><mn>1<\/mn> <\/mrow>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mn>1<\/mn>  <\/mtd><\/mtr> <\/mtable>  <\/mrow>  <mo>)<\/mo><\/mrow><mo>,<\/mo><mrow>  <mo>(<\/mo>  <mrow> <mtable><mtr>  <mtd> <mn>3<\/mn>  <\/mtd>  <mtd> <mn>2<\/mn>  <\/mtd>  <mtd> <mrow><mi>\u2212<\/mi><mn>1<\/mn> <\/mrow>  <\/mtd><\/mtr><mtr>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mfrac><mn>11<\/mn><mn>3<\/mn> <\/mfrac>  <\/mtd>  <mtd> <mfrac><mn>20<\/mn><mn>3<\/mn> <\/mfrac>  <\/mtd><\/mtr><mtr>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mn>0<\/mn>  <\/mtd>  <mtd> <mn>6<\/mn>  <\/mtd><\/mtr> <\/mtable>  <\/mrow>  <mo>)<\/mo><\/mrow><mo>]<\/mo> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math>\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<h3>Bibliograf\u00eda<\/h3>\n<ul>\n<li>Arriaza G\u00f3mez A. J., del \u00c1guila Garrido L., Rambla Barreno F., Redondo Neble M. V., Rodr\u00edguez Galv\u00e1n J. R., Viglialoro G. Manual de pr\u00e1cticas de Matem\u00e1ticas con M\u00e1xima. C\u00e1diz: Editorial UCA; 2015.<\/li>\n<\/ul>\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> Sea la matriz \\(A\\)=[[1,-1,1,2],[0,0,3,-1],[4,2,0,2],[1,0,3,-1]], \u00bfcu\u00e1l es el mayor valor de sus menores principales? <\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>35<\/li>\n<li>28<\/li>\n<li>20<\/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<p><strong>C.)<\/strong><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGmenorprin01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\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 transformaciones&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[7],"class_list":["post-171","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\/171","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=171"}],"version-history":[{"count":25,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/171\/revisions"}],"predecessor-version":[{"id":185,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/171\/revisions\/185"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=171"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=171"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=171"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}