{"id":199,"date":"2026-10-05T15:25:42","date_gmt":"2026-10-05T13:25:42","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=199"},"modified":"2026-09-23T20:36:16","modified_gmt":"2026-09-23T18:36:16","slug":"mathbio-algebra-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=199","title":{"rendered":"MathBio: \u00c1lgebra con Maxima"},"content":{"rendered":"<p>Hoy nos iniciamos en un sistema para la manipulaci\u00f3n de expresiones simb\u00f3licas y num\u00e9ricas, Maxima. Un herramienta inform\u00e1tica que nos ayudar\u00e1 a resolver problemas de la asignatura de forma sencilla y aplicada.<\/p>\n<p>Para comenzar nos iniciaremos en la definici\u00f3n de vectores y matrices, y las operaciones que podemos hacer con ellos.<\/p>\n<h2>Vectores<\/h2>\n<p>Un vector se define utilizando [] y los elementos del vector separados por comas. Con los vectores podemos hacer las operaciones b\u00e1sicas de suma y multiplicaci\u00f3n por escalar.<\/p>\n<p><code><br \/>\nv:[1,2,3];<br \/>\n3.v+(-2).u;<\/p>\n<p><\/code><\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong>  Sean los vectores \\(\\mathbf{u}=[2,-1,5,0]\\), \\(\\mathbf{v}=[4,3,1,-1]\\) y \\(\\mathbf{w}=[-6,2,0,3]\\). Si \\(\\mathbf{x}+\\mathbf{v}+3\\mathbf{w}=2\\mathbf{u}\\), \u00bfcu\u00e1nto suman las coordenadas de \\(\\mathbf{x}\\)? <\/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<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/ejrALGvectores01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sean los vectores \\(u\\)=[2,-1], \\(v\\)=[1,1] y \\(w\\)=[-1,3], \u00bfcu\u00e1nto es \\(\\textbf{proy}_{w}(u)\\bullet\\textbf{proy}_{w}(v)\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3dx() {\n  var htmlShow3dx = document.getElementById(\"html-show3dx\");\n  if (htmlShow3dx.style.display === \"none\") {\n    htmlShow3dx.style.display = \"block\";\n  } else {\n    htmlShow3dx.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv3dx()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3dx\" 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\">(%i6) <\/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_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">w<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/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\">uw<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">w<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">vw<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">w<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">ww<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">w<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">w<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }-5\\]<\/p>\n<p>\\[\\operatorname{ }2\\]<\/p>\n<p>\\[\\operatorname{ }10\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_operator\">(<\/span><span class=\"code_variable\">uw<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">ww<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">w<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">vw<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_variable\">ww<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">w<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }-1\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Matrices<\/h2>\n<p>Si queremos utilizar matrices nos bastar\u00e1 con definirla mediante <strong>matrix()<\/strong>. Las filas de definimos como vectores:<\/p>\n<p><code><br \/>\nA:matrix([1,2,3],[4,5,6]);<br \/>\nB:matrix([1,2],[3,4],[5,6]);<\/p>\n<p><\/code><\/p>\n<p>La primera, A, ser\u00eda una matriz de 2&#215;3, B ser\u00eda una matriz de 3&#215;2. La manera de acceder a los elementos es mediante A[i,j].<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(A\\)=[[4,-1,6],[2,1,6],[2,-1,8]] y \\(B\\)=[[0,-1,5],[1,6,2],[1,8,0]]. \u00bfCu\u00e1l es la suma de los elementos de la diagonal principal de \\(2A-3B\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv23() {\n  var htmlShow23 = document.getElementById(\"html-show23\");\n  if (htmlShow23.style.display === \"none\") {\n    htmlShow23.style.display = \"block\";\n  } else {\n    htmlShow23.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv23()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show23\" 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\">(%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_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">6<\/span>],[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">6<\/span>],[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">8<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">B<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">0<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">5<\/span>],[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">6<\/span>,<span class=\"code_number\">2<\/span>],[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">8<\/span>,<span class=\"code_number\">0<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">C<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">*<\/span><span class=\"code_variable\">B<\/span><span class=\"code_endofline\">;<\/span><br \/><span class=\"code_variable\">C<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">+<\/span><span class=\"code_variable\">C<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">2<\/span>]<span class=\"code_operator\">+<\/span><span class=\"code_variable\">C<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">3<\/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=\"center\"> <mtd><mtext>(C)<\/mtext> <\/mtd> <mtd><mrow>  <mo>[<\/mo>  <mrow> <mtable><mtr>  <mtd> <mn>8<\/mn>  <\/mtd>  <mtd> <mn>1<\/mn>  <\/mtd>  <mtd> <mrow><mi>\u2212<\/mi><mn>3<\/mn> <\/mrow>  <\/mtd><\/mtr><mtr>  <mtd> <mn>1<\/mn>  <\/mtd>  <mtd> <mrow><mi>\u2212<\/mi><mn>16<\/mn> <\/mrow>  <\/mtd>  <mtd> <mn>6<\/mn>  <\/mtd><\/mtr><mtr>  <mtd> <mn>1<\/mn>  <\/mtd>  <mtd> <mrow><mi>\u2212<\/mi><mn>26<\/mn> <\/mrow>  <\/mtd>  <mtd> <mn>16<\/mn>  <\/mtd><\/mtr> <\/mtable>  <\/mrow>  <mo>]<\/mo><\/mrow> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable>  <mlabeledtr columnalign=\"center\"> <mtd><mtext>(%o4) <\/mtext> <\/mtd> <mtd><mn>8<\/mn> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<p>Otros comandos:<\/p>\n<ul>\n<li><strong>col<\/strong>((<em>Matriz<\/em>,<em>N\u00famColumna<\/em>)): Recupera la columna <em>N\u00famColumna<\/em>. <\/li>\n<li><strong>row<\/strong>((<em>Matriz<\/em>,<em>N\u00famFila<\/em>)): Recupera la fila <em>N\u00famFila<\/em>. <\/li>\n<li><strong>submatrix<\/strong>(\\(i_1,i_2,\\ldots,i_p\\), <em>Matriz<\/em>,\\(j_1,j_2,\\ldots,j_q\\)): Elimina de la <em>Matriz <\/em>las filas cuyos n\u00fameros son \\(i_1,i_2,\\ldots,i_p\\) y las columnas cuyos n\u00fameros son \\(j_1,j_2,\\ldots,j_q\\). No es preciso que est\u00e9n ambas: pueden eliminarse \u00fanicamente filas o columnas.<\/li>\n<li><strong>addrow<\/strong>(<em>Matriz<\/em>, \\(v_1, \\ldots, v_p\\)): A\u00f1ade en la base de <em>Matriz <\/em>las filas dadas por vectores (o matrices) \\(v_1, \\ldots, v_p\\). Las longitudes deben ser concordantes.<\/li>\n<li><strong>addcol<\/strong>(<em>Matriz<\/em>, \\(v_1, \\ldots, v_p\\)): A\u00f1ade en la base de <em>Matriz <\/em>las filas dadas por vectores (o matrices) \\(v_1, \\ldots, v_p\\). Las longitudes deben ser concordantes.<\/li>\n<li><strong>matrix_size<\/strong>(<em>Matriz<\/em>): Proporciona las dimensiones de la matriz.<\/li>\n<li><strong>transpose<\/strong>(<em>Matriz<\/em>): Proporciona la matriz traspuesta de <em>Matriz<\/em>.<\/li>\n<\/ul>\n<p>Las operaciones con matrices son semejantes a las utilizadas con los vectores.<\/p>\n<p>Algunas matrices interesantes:<\/p>\n<ul>\n<li><strong>diagmatrix<\/strong>(<em>N\u00famero<\/em>,<em>Valor<\/em>): Genera una matriz cuadrada diagonal cuyo tama\u00f1o se establece mediante el valor de <em>N\u00famero <\/em> y en la que todos los elementos de la diagonal tienen el mismo <em>Valor<\/em>. <\/li>\n<li><strong>diag_matrix<\/strong>(\\(a_1,a_2,\\ldots,a_n\\)): Genera una matriz diagonal cuadrada con \\(a_1,a_2,\\ldots,a_n\\) en la diagonal. <\/li>\n<li><strong>ident<\/strong>(<em>N\u00famero<\/em>): Genera la matriz identidad (cuadrada) cuyo tama\u00f1o viene dado por el valor <em>N\u00famero<\/em>; es un caso particular del anterior.<\/li>\n<li><strong>zeromatrix<\/strong>(<em>n<\/em>,<em>m<\/em>): Genera la matriz de <em>n<\/em> filas y <em>m<\/em> columnas en la que todos sus elementos son ceros.<\/li>\n<\/ul>\n<p>Uno de los ejercicios m\u00e1s comunes que realizaremos ser\u00e1 el c\u00e1lculo de rango, determinantes, menores e inversa de una matriz. Primero aprenderemos a realizarlo mediante operaciones elementales, consiguiendo una matriz escalonada o una matriz triangular. No obstante, tenemos los comandos que nos los proporcionan:<\/p>\n<ul>\n<li><strong>rank<\/strong>(\\(M\\)): dada la matriz \\(M\\) nos devuelve su rango.<\/li>\n<li><strong>determinant<\/strong>(\\(M\\)): dada la matriz \\(M\\) nos devuelve su determinante.<\/li>\n<li><strong>mat_trace<\/strong>(\\(M\\)): dada la matriz \\(M\\) nos devuelve su traza.<\/li>\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<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<li><strong>ratsimp<\/strong>(<em>expr<\/em>): Simplifica la expresi\u00f3n <em>expr<\/em> y todas sus subexpresiones, incluyendo los argumentos de funciones no racionales. <\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Cu\u00e1l es el valor de x para que el rango de la matriz sea 2  \\[\\begin{bmatrix}<br \/>\n5 &#038; -5 &#038; -6\\\\<br \/>\n-5 &#038; 3 &#038; -1 \\\\<br \/>\n0 &#038; x &#038;7<br \/>\n\\end{bmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv71() {\n  var htmlShow71 = document.getElementById(\"html-show71\");\n  if (htmlShow71.style.display === \"none\") {\n    htmlShow71.style.display = \"block\";\n  } else {\n    htmlShow71.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv71()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show71\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">ratsimp<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}5 &amp; -5 &amp; -6\\\\-5 &amp; 3 &amp; -1\\\\0 &amp; x &amp; 7\\end{bmatrix}\\]<\/p>\n<p>\\[\\operatorname{ }35 x-70\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ x=2\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \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 showHtmlDiv71e() {\n  var htmlShow71e = document.getElementById(\"html-show71e\");\n  if (htmlShow71e.style.display === \"none\") {\n    htmlShow71e.style.display = \"block\";\n  } else {\n    htmlShow71e.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv71e()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show71e\" 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>  \u00bfCu\u00e1l es la ecuaci\u00f3n impl\u00edcita del plano que pasa por los puntos \\(P(1,2,3)\\), \\(Q(-1,0,2)\\) y \\(R(4,-2,0)\\)?  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv31() {\n  var htmlShow31 = document.getElementById(\"html-show31\");\n  if (htmlShow31.style.display === \"none\") {\n    htmlShow31.style.display = \"block\";\n  } else {\n    htmlShow31.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv31()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show31\" 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\">(%i4) <\/span>  <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">0<\/span>,<span class=\"code_number\">2<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">0<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">ratsimp<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">matrix<\/span>([<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>]<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span>,<span class=\"code_variable\">Q<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span>,<span class=\"code_variable\">R<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span>))<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/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>(%o4) <\/mtext> <\/mtd> <mtd><mn>14<\/mn><mo>\u2062<\/mo><mi>z<\/mi><mi>\u2212<\/mi><mn>9<\/mn><mo>\u2062<\/mo><mi>y<\/mi><mo>+<\/mo><mn>2<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mi>\u2212<\/mi><mn>26<\/mn><mi>=<\/mi><mn>0<\/mn> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Justificar si el punto S(2,4,-13) es coplanario con los puntos P(1,-3,-1), Q(2,-2,1) y R(3,2,-4)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv6q() {\n  var htmlShow6q = document.getElementById(\"html-show6q\");\n  if (htmlShow6q.style.display === \"none\") {\n    htmlShow6q.style.display = \"block\";\n  } else {\n    htmlShow6q.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv6q()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show6q\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Puntos Coplanarios. Ej.2 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/9g6oZ805A5Y?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> \u00bfCu\u00e1l es el valor del producto escalar del vector \\([1,-2,3]\\) por el vector unitario normal de la ecuaci\u00f3n impl\u00edcita del plano que pasa por los puntos \\(P(1,-3,-1)\\), \\(Q(2,-2,1)\\) y \\(R(3,2,-4)\\)?  <\/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\">(%i6) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">;<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">;<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">4<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">pq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">pr<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">rat<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">matrix<\/span>([<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>]<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span>,<span class=\"code_variable\">pq<\/span>,<span class=\"code_variable\">pr<\/span>))<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/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>(P)<\/mtext> <\/mtd> <mtd><mo>[<\/mo><mn>1<\/mn><mo>,<\/mo><mi>\u2212<\/mi><mn>3<\/mn><mo>,<\/mo><mi>\u2212<\/mi><mn>1<\/mn><mo>]<\/mo> <\/mtd><\/mlabeledtr><\/mtable> <\/math> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(Q)<\/mtext> <\/mtd> <mtd><mo>[<\/mo><mn>2<\/mn><mo>,<\/mo><mi>\u2212<\/mi><mn>2<\/mn><mo>,<\/mo><mn>1<\/mn><mo>]<\/mo> <\/mtd><\/mlabeledtr><\/mtable> <\/math> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(%o6)\/R\/ <\/mtext> <\/mtd> <mtd><mn>3<\/mn><mo>\u2062<\/mo><mi>z<\/mi><mo>+<\/mo><mn>7<\/mn><mo>\u2062<\/mo><mi>y<\/mi><mi>\u2212<\/mi><mn>13<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>+<\/mo><mn>37<\/mn><mi>=<\/mi><mn>0<\/mn> <\/mtd><\/mlabeledtr><\/mtable> <\/math> <!-- 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\">vn<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">13<\/span>,<span class=\"code_number\">7<\/span>,<span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><br \/>(<span class=\"code_variable\">vn<\/span><span class=\"code_number\">.[<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>])<span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">vn<\/span>.<span class=\"code_variable\">vn<\/span>),<span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span> <\/span><\/td>\n<\/tr>\n<\/table>\n<p> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"> <mtd><mtext>(%o8) <\/mtext> <\/mtd> <mtd><mi>\u2212<\/mi><mn>1.19470196087995<\/mn> <\/mtd><\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \u00bfCu\u00e1l es la norma del vector perpendicular a los vectores \\(\\vec{v}:[1,-2,3]\\) y \\(\\vec{u}:[3,1,-1]\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3a() {\n  var htmlShow3a = document.getElementById(\"html-show3a\");\n  if (htmlShow3a.style.display === \"none\") {\n    htmlShow3a.style.display = \"block\";\n  } else {\n    htmlShow3a.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv3a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3a\" style=\"display: none;\">\n <!-- Code cell -->  <\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"> <span class=\"prompt\">(%i4) <\/span>  <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">v<\/span>,<span class=\"code_variable\">u<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">vxu<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">minor<\/span>(<span class=\"code_variable\">A<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>)),<br \/>    <span class=\"code_operator\">&#8211;<\/span><span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">minor<\/span>(<span class=\"code_variable\">A<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>)),<br \/>    <span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">minor<\/span>(<span class=\"code_variable\">A<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">3<\/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=\"center\"> <mtd><mtext>(vxu)<\/mtext> <\/mtd> <mtd><mo>[<\/mo><mi>\u2212<\/mi><mn>1<\/mn><mo>,<\/mo><mn>10<\/mn><mo>,<\/mo><mn>7<\/mn><mo>]<\/mo> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math> <!-- 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\">sqrt<\/span>(<span class=\"code_variable\">vxu<\/span>.<span class=\"code_variable\">vxu<\/span>),<span class=\"code_variable\">numer<\/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=\"center\"> <mtd><mtext>(%o5) <\/mtext> <\/mtd> <mtd><mn>12.24744871391589<\/mn> <\/mtd>  <\/mlabeledtr><\/mtable> <\/math>\n<\/div>\n<h2>Escalonado de matrices en maxima<\/h2>\n<p>El pasado d\u00eda vimos como realizar transformaciones elementales para encontrar una matriz escalonada de cualquier matriz. Este proceso nos ayuda en el c\u00e1lculo de la inversa de una matriz. Estas operaciones son f\u00e1ciles con maxima utilizando estos comandos:<\/p>\n<ul>\n<li><strong>rowswap<\/strong>(\\(M\\), i, j): dada la matriz \\(M\\) nos devuelve la misma donde la se han intercambiado las filas <em>i<\/em> y <em>j<\/em>, \\(f_i\\leftrightarrow f_j\\).<\/li>\n<li><strong>columnswap<\/strong>(\\(M\\), i, j): dada la matriz \\(M\\) nos devuelve la misma donde la se han intercambiado las columnas <em>i<\/em> y <em>j<\/em>, \\(c_i\\leftrightarrow c_j\\).<\/li>\n<li><strong>rowop<\/strong>(\\(M\\), i, j, \\(\\alpha\\)): dada la matriz \\(M\\) nos devuelve la misma donde la \\(f_i\\leftarrow f_i-\\alpha f_j\\).<\/li>\n<li><strong>columnop<\/strong>(\\(M\\), i, j, \\(\\alpha\\)): dada la matriz \\(M\\) nos devuelve la misma donde la \\(c_i\\leftarrow c_i-\\alpha c_j\\).<\/li>\n<\/ul>\n<p>Con estos comandos podemos realizar las operaciones elementales que tratamos en clases anteriores. Sin embargo, una de las operaciones tiene un procedimiento m\u00e1s delicado: la multiplicaci\u00f3n de una fila por un escalar. Imaginemos que el elemento \\(a_{ic}=\\gamma\\) de una matriz queremos que su valor sea \\(\\beta\\). Necesitamos saber qu\u00e9 escalar \\(\\alpha\\) debemos multiplicar a la fila \\(f_i\\) para que el comando <strong>rowop<\/strong>(\\(M\\), i, i, \\(\\alpha\\)) transforme \\(a_{ic}=\\gamma\\) en \\(a_{ic}=\\beta\\). Luego \\[\\beta f_i={\\gamma}f_i \\ -\\ {\\gamma}{\\alpha}f_i\\to \\alpha=1-\\frac{\\beta}{\\gamma}.\\]<br \/>\nDe esta forma, solo necesitamos sustituir para encontrar el \\(\\alpha\\) apropiado que nos proporcione \\(\\beta\\).<\/p>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Calcular la inversa de la matriz \\(A\\)=[[2,0,0],[1,2,0],[2,-1,3]]? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3b5() {\n  var htmlShow3b5 = document.getElementById(\"html-show3b5\");\n  if (htmlShow3b5.style.display === \"none\") {\n    htmlShow3b5.style.display = \"block\";\n  } else {\n    htmlShow3b5.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv3b5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3b5\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">Definamos la matriz y la ampliada con la que trabajaremos<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">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\">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_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\">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\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">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; 2 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0 &amp; 1 &amp; 2 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 2 &amp; -1 &amp; 3\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Para trabajar m\u00e1s c\u00f3modo hacemos que el primer elemento de X[1,4] sea 1, para ello es suficiente con intercambiar las filas 1 y 2<\/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\">(%i3) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/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\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 1 &amp; 2 &amp; 0\\\\1 &amp; 0 &amp; 0 &amp; 2 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 2 &amp; -1 &amp; 3\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora podemos hacer ceros los elementos por debajo del X1[1,4]:<\/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\">X2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/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}0 &amp; 1 &amp; 0 &amp; 1 &amp; 2 &amp; 0\\\\1 &amp; -2 &amp; 0 &amp; 0 &amp; -4 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 2 &amp; -1 &amp; 3\\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\">X3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">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}0 &amp; 1 &amp; 0 &amp; 1 &amp; 2 &amp; 0\\\\1 &amp; -2 &amp; 0 &amp; 0 &amp; -4 &amp; 0\\\\0 &amp; -2 &amp; 1 &amp; 0 &amp; -5 &amp; 3\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El siguiente proceso es que el elemento X3[5,2] sea un 1. Para coseguirlo tenemos diferentes m\u00e9todos, en este caso el m\u00e1s simple es operar con la fila siguiente:<\/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_variable\">X4<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 1 &amp; 2 &amp; 0\\\\1 &amp; 0 &amp; -1 &amp; 0 &amp; 1 &amp; -3\\\\0 &amp; -2 &amp; 1 &amp; 0 &amp; -5 &amp; 3\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora hacemos cero el elemento X4[3,5]<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X5<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X4<\/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\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 1 &amp; 2 &amp; 0\\\\1 &amp; 0 &amp; -1 &amp; 0 &amp; 1 &amp; -3\\\\5 &amp; -2 &amp; -4 &amp; 0 &amp; 0 &amp; -12\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El siguiente paso es conseguir que el elemento X5[3,6] sea un 1, para ello operamos con el escalar 1-1\/(-12)<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X6<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/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\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">12<\/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}0 &amp; 1 &amp; 0 &amp; 1 &amp; 2 &amp; 0\\\\1 &amp; 0 &amp; -1 &amp; 0 &amp; 1 &amp; -3\\\\-\\frac{5}{12} &amp; \\frac{1}{6} &amp; \\frac{1}{3} &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora solo nos resta escalonar superiormente:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X7<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X6<\/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_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 1 &amp; 2 &amp; 0\\\\-\\frac{1}{4} &amp; \\frac{1}{2} &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\-\\frac{5}{12} &amp; \\frac{1}{6} &amp; \\frac{1}{3} &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X8<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X7<\/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\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}\\frac{1}{2} &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\-\\frac{1}{4} &amp; \\frac{1}{2} &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\-\\frac{5}{12} &amp; \\frac{1}{6} &amp; \\frac{1}{3} &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ya hemos conseguido que la matriz ampliada [I|A] se transforme en [B|I], por tanto la matriz B ser\u00e1 la inversa de A:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i11)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">invA<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X8<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}\\frac{1}{2} &amp; 0 &amp; 0\\\\-\\frac{1}{4} &amp; \\frac{1}{2} &amp; 0\\\\-\\frac{5}{12} &amp; \\frac{1}{6} &amp; \\frac{1}{3}\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Verifiqu\u00e9moslo<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i13)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">invA<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">invA<\/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>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Calcular la inversa de la matriz \\(A\\)=[[3,0,-1,1], [1,1,2,-1],[0,1,-1,0], [1,2,0,-1]]? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3b() {\n  var htmlShow3b = document.getElementById(\"html-show3b\");\n  if (htmlShow3b.style.display === \"none\") {\n    htmlShow3b.style.display = \"block\";\n  } else {\n    htmlShow3b.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv3b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3b\" 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\">3<\/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\">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_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\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/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\">1<\/span><span class=\"code_endofline\">,<\/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\">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\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">A<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 3 &amp; 0 &amp; -1 &amp; 1\\\\0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; 0 &amp; -1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Veamos c\u00f3mo escalonamos las columnas de 5 a 8:<\/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\">(%i3) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/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\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\1 &amp; 0 &amp; 0 &amp; 0 &amp; 3 &amp; 0 &amp; -1 &amp; 1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &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\">(%i4) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X2<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">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><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\1 &amp; -3 &amp; 0 &amp; 0 &amp; 0 &amp; -3 &amp; -7 &amp; 4\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; 0 &amp; -1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X3<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\1 &amp; -3 &amp; 0 &amp; 0 &amp; 0 &amp; -3 &amp; -7 &amp; 4\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; -1 &amp; 0 &amp; 1 &amp; 0 &amp; 1 &amp; -2 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X4<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X3<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\1 &amp; -3 &amp; 0 &amp; 0 &amp; 0 &amp; -3 &amp; -7 &amp; 4\\\\0 &amp; -1 &amp; 0 &amp; 1 &amp; 0 &amp; 1 &amp; -2 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X5<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X4<\/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_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\1 &amp; -3 &amp; 3 &amp; 0 &amp; 0 &amp; 0 &amp; -10 &amp; 4\\\\0 &amp; -1 &amp; 0 &amp; 1 &amp; 0 &amp; 1 &amp; -2 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X6<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\1 &amp; -3 &amp; 3 &amp; 0 &amp; 0 &amp; 0 &amp; -10 &amp; 4\\\\0 &amp; -1 &amp; -1 &amp; 1 &amp; 0 &amp; 0 &amp; -1 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X7<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; -1 &amp; -1 &amp; 1 &amp; 0 &amp; 0 &amp; -1 &amp; 0\\\\1 &amp; -3 &amp; 3 &amp; 0 &amp; 0 &amp; 0 &amp; -10 &amp; 4\\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\">X8<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/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_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; 1 &amp; 1 &amp; -1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\1 &amp; -3 &amp; 3 &amp; 0 &amp; 0 &amp; 0 &amp; -10 &amp; 4\\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\">X9<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X8<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">10<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; 1 &amp; 1 &amp; -1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\1 &amp; 7 &amp; 13 &amp; -10 &amp; 0 &amp; 0 &amp; 0 &amp; 4\\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\">X10<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X9<\/span><span class=\"code_endofline\">,<\/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\">1<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\/<\/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>\\[\\operatorname{ }\\begin{bmatrix}0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; 1 &amp; 1 &amp; -1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\\\frac{1}{4} &amp; \\frac{7}{4} &amp; \\frac{13}{4} &amp; -\\frac{5}{2} &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Ahora hacemos el proceso de triangularizar la parte de arriba.<\/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\">(%i13)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X11<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X10<\/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\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}\\frac{1}{4} &amp; \\frac{11}{4} &amp; \\frac{13}{4} &amp; -\\frac{5}{2} &amp; 1 &amp; 1 &amp; 2 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; 1 &amp; -1 &amp; 0\\\\0 &amp; 1 &amp; 1 &amp; -1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\\\frac{1}{4} &amp; \\frac{7}{4} &amp; \\frac{13}{4} &amp; -\\frac{5}{2} &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i14)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X12<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X11<\/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_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}\\frac{1}{4} &amp; \\frac{11}{4} &amp; \\frac{13}{4} &amp; -\\frac{5}{2} &amp; 1 &amp; 1 &amp; 2 &amp; 0\\\\0 &amp; 1 &amp; 2 &amp; -1 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 1 &amp; -1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\\\frac{1}{4} &amp; \\frac{7}{4} &amp; \\frac{13}{4} &amp; -\\frac{5}{2} &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i15)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X13<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X12<\/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_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}\\frac{1}{4} &amp; \\frac{3}{4} &amp; \\frac{5}{4} &amp; -\\frac{1}{2} &amp; 1 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 2 &amp; -1 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 1 &amp; -1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\\\frac{1}{4} &amp; \\frac{7}{4} &amp; \\frac{13}{4} &amp; -\\frac{5}{2} &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i16)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X14<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X13<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}\\frac{1}{4} &amp; -\\frac{1}{4} &amp; -\\frac{3}{4} &amp; \\frac{1}{2} &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 2 &amp; -1 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 1 &amp; -1 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\\\frac{1}{4} &amp; \\frac{7}{4} &amp; \\frac{13}{4} &amp; -\\frac{5}{2} &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Por tanto, la inversa ser\u00e1<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i17)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X14<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">7<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">8<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\operatorname{ }\\begin{bmatrix}\\frac{1}{4} &amp; -\\frac{1}{4} &amp; -\\frac{3}{4} &amp; \\frac{1}{2}\\\\0 &amp; 1 &amp; 2 &amp; -1\\\\0 &amp; 1 &amp; 1 &amp; -1\\\\\\frac{1}{4} &amp; \\frac{7}{4} &amp; \\frac{13}{4} &amp; -\\frac{5}{2}\\end{bmatrix}\\]<\/p>\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 [[1,2,-3],[-2,0,4],[0,4,-2],[-2,-4,\\(a\\)]], \u00bfcu\u00e1l es el valor de \\(a\\) para que el rango de la matriz sea par? <\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>3<\/li>\n<li>6<\/li>\n<li>9<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv(id) {\n  var htmlShow = document.getElementById(id);\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<p><em>La soluci\u00f3n se mostrar\u00e1 m\u00e1s adelante.<\/em><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Hoy nos iniciamos en un sistema para la manipulaci\u00f3n de expresiones simb\u00f3licas y num\u00e9ricas, Maxima. Un herramienta inform\u00e1tica que nos ayudar\u00e1 a resolver problemas de la asignatura de forma sencilla y aplicada&#8230;.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[21],"class_list":["post-199","post","type-post","status-publish","format-standard","hentry","category-mathbio","tag-practicas-mathbio"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/199","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=199"}],"version-history":[{"count":9,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/199\/revisions"}],"predecessor-version":[{"id":256,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/199\/revisions\/256"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=199"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=199"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=199"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}