{"id":167,"date":"2025-10-03T10:16:49","date_gmt":"2025-10-03T08:16:49","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=167"},"modified":"2025-10-03T19:23:41","modified_gmt":"2025-10-03T17:23:41","slug":"alg-matrices-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=167","title":{"rendered":"ALG: Matrices con Maxima"},"content":{"rendered":"<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<blockquote>\n<p><strong>Ejercicio:<\/strong> Definir las matrices  \\(A=\\begin{bmatrix}1 &#038; 2 &#038; 3\\\\<br \/>\n 4 &#038; 5 &#038; 6\\end{bmatrix}\\) y \\(B=\\begin{bmatrix}1 &#038; 2 \\\\ 3&#038; 4 \\\\ 5 &#038; 6\\end{bmatrix}\\). <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv32() {\n  var htmlShow32 = document.getElementById(\"html-show32\");\n  if (htmlShow32.style.display === \"none\") {\n    htmlShow32.style.display = \"block\";\n  } else {\n    htmlShow32.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv32()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show32\" style=\"display: none;\">\n   <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i2)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n          <span class=\"input\">           <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\">2<\/span>,            <span class=\"code_number\">3<\/span>],[            <span class=\"code_number\">4<\/span>,            <span class=\"code_number\">5<\/span>,            <span class=\"code_number\">6<\/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\">1<\/span>,            <span class=\"code_number\">2<\/span>],[            <span class=\"code_number\">3<\/span>,            <span class=\"code_number\">4<\/span>],[            <span class=\"code_number\">5<\/span>,            <span class=\"code_number\">6<\/span>])            <span class=\"code_endofline\">;<\/span>          <\/span>\n        <\/td>\n<\/tr>\n<\/table>\n<p>    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(A)<\/mtext>\n          <\/mtd>\n          <mtd>\n            <mrow>\n              <mo>[<\/mo>\n              <mrow>\n                <mtable>\n                  <mtr>\n                    <mtd>\n                      <mn>1<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>2<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>3<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>4<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>5<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>6<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                <\/mtable>\n              <\/mrow>\n              <mo>]<\/mo>\n            <\/mrow>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math><br \/>\n    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(B)<\/mtext>\n          <\/mtd>\n          <mtd>\n            <mrow>\n              <mo>[<\/mo>\n              <mrow>\n                <mtable>\n                  <mtr>\n                    <mtd>\n                      <mn>1<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>2<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>3<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>4<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>5<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>6<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                <\/mtable>\n              <\/mrow>\n              <mo>]<\/mo>\n            <\/mrow>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math>\n<\/div>\n<hr \/>\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<p>Los operadores anteriores comentados para vectores funcionan igual en matrices. En el caso de elevar una matriz a una potencia tendr\u00edamos que utilizar &quot;<strong>^^<\/strong>&quot;. <\/p>\n<p>Otra forma de acceder a submatrices es con los 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<li><strong>mat_trace<\/strong>(\\(M\\)): dada la matriz \\(M\\) nos devuelve su traza.<\/li>\n<\/ul>\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<blockquote>\n<p><strong>Ejercicio:<\/strong> Construir la matriz \\[\\begin{pmatrix}2 &#038; 2 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n2 &#038; 2 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; -1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 3 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 3 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 &#038; 3 &#038; 0 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 4 &#038; 4 &#038; 4 &#038; 4\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 4 &#038; 4 &#038; 4 &#038; 4\\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 4 &#038; 4 &#038; 4 &#038; 4\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 4 &#038; 4 &#038; 4 &#038; 4\\end{pmatrix}\\]  <\/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><br \/>\n<button onclick=\"showHtmlDiv3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i7)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n<span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/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_operator\">+<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">2<\/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\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/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_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">X1<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">zeromatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">+<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">8<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">8<\/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_number\">1<\/span><span class=\"code_endofline\">$<\/span> <\/span><\/td>\n<\/tr>\n<\/table>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Escribir en forma matricial el sistema \\[\\begin{matrix}x+2y=10\\\\ 2x-2y=4\\\\ 3x+5y=26\\end{matrix}\\]  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4() {\n  var htmlShow4 = document.getElementById(\"html-show4\");\n  if (htmlShow4.style.display === \"none\") {\n    htmlShow4.style.display = \"block\";\n  } else {\n    htmlShow4.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i4)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n<span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/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_variable\">y<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">10<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">26<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\">  <br \/><\/span><span class=\"code_function\">print<\/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_endofline\">,<\/span><span class=\"code_string\">&quot;=&quot;<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">b<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span> <\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{pmatrix}1 &amp; 2\\\\ 2 &amp; -2\\\\ 3 &amp; 5\\end{pmatrix} \\begin{pmatrix}x\\\\ y\\end{pmatrix}=\\begin{pmatrix}10\\\\ 4\\\\ 26\\end{pmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dadas \\(A\\)=[[1,-3],[2,2],[4,1]] y \\(B\\)=[[-1,-7], [4,-5]], Si \\(C=B^t\\cdot A^t\\), \u00bfcu\u00e1nto es \\(c_{12}\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv5() {\n  var htmlShow5 = document.getElementById(\"html-show5\");\n  if (htmlShow5.style.display === \"none\") {\n    htmlShow5.style.display = \"block\";\n  } else {\n    htmlShow5.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table >\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">    <span class=\"prompt\">(%i4)\t  <\/span>  <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>],[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">2<\/span>],[<span class=\"code_number\">4<\/span>,<span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">B<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">matrix<\/span>([<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">7<\/span>], [<span class=\"code_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">5<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">C<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span>(<span class=\"code_variable\">A<\/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\">2<\/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\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(C)<\/mtext>\n          <\/mtd>\n          <mtd>\n            <mrow>\n              <mo>[<\/mo>\n              <mrow>\n                <mtable>\n                  <mtr>\n                    <mtd>\n                      <mrow>\n                        <mi>\u2212<\/mi>\n                        <mn>13<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                    <mtd>\n                      <mn>6<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mn>0<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>8<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mrow>\n                        <mi>\u2212<\/mi>\n                        <mn>24<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                    <mtd>\n                      <mrow>\n                        <mi>\u2212<\/mi>\n                        <mn>33<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                  <\/mtr>\n                <\/mtable>\n              <\/mrow>\n              <mo>]<\/mo>\n            <\/mrow>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math><br \/>\n    <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(%o4) <\/mtext>\n          <\/mtd>\n          <mtd>\n            <mn>6<\/mn>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea \\(A=\\begin{bmatrix} 1 &#038; 3\\\\  \\alpha &#038; 1 \\end{bmatrix}\\), \u00bfcu\u00e1l es el valor de \\(\\alpha\\) para el cual A es una ra\u00edz del polinomio \\(f(x)=x^2-2x-8\\)?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv6() {\n  var htmlShow6 = document.getElementById(\"html-show6\");\n  if (htmlShow6.style.display === \"none\") {\n    htmlShow6.style.display = \"block\";\n  } else {\n    htmlShow6.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<button onclick=\"showHtmlDiv6()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show6\" style=\"display: none;\">\n    <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\">\n          <span class=\"prompt\">(%i2)<br \/>\n  <\/span>\n        <\/td>\n<td style=\"vertical-align: top;padding: 1mm;\">\n<span class=\"input\">  <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\">3<\/span>], [  <span class=\"code_variable\">x<\/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_operator\">^<\/span>  <span class=\"code_number\">2<\/span>  <span class=\"code_operator\">&#8211;<\/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\">8<\/span>  <span class=\"code_operator\">*<\/span>  <span class=\"code_function\">ident<\/span>(  <span class=\"code_number\">2<\/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\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(%o2) <\/mtext>\n          <\/mtd>\n          <mtd>\n            <mrow>\n              <mo>[<\/mo>\n              <mrow>\n                <mtable>\n                  <mtr>\n                    <mtd>\n                      <mrow>\n                        <mn>3<\/mn>\n                        <mo>\u2062<\/mo>\n                        <mi>x<\/mi>\n                        <mi>\u2212<\/mi>\n                        <mn>9<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                    <mtd>\n                      <mn>0<\/mn>\n                    <\/mtd>\n                  <\/mtr>\n                  <mtr>\n                    <mtd>\n                      <mn>0<\/mn>\n                    <\/mtd>\n                    <mtd>\n                      <mrow>\n                        <mn>3<\/mn>\n                        <mo>\u2062<\/mo>\n                        <mi>x<\/mi>\n                        <mi>\u2212<\/mi>\n                        <mn>9<\/mn>\n                      <\/mrow>\n                    <\/mtd>\n                  <\/mtr>\n                <\/mtable>\n              <\/mrow>\n              <mo>]<\/mo>\n            <\/mrow>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math><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\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">%<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>],<span class=\"code_variable\">x<\/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\">\n      <mtable>\n        <mlabeledtr columnalign=\"center\">\n          <mtd>\n            <mtext>(%o3) <\/mtext>\n          <\/mtd>\n          <mtd>\n            <mo>[<\/mo>\n            <mi>x<\/mi>\n            <mi>=<\/mi>\n            <mn>3<\/mn>\n            <mo>]<\/mo>\n          <\/mtd>\n        <\/mlabeledtr>\n      <\/mtable>\n    <\/math><\/p>\n<\/div>\n<hr \/>\n<h2>Operaciones con matrices en maxima<\/h2>\n<p>El pasado d\u00eda vimos como realizar transformaciones elementales para encontrar una matriz escalonada de cualquier 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<\/ul>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dar las matrices de paso, por la izquierda y por la derecha, que verifica la semejanza \\[\\begin{bmatrix}<br \/>\n0 &#038; 0 &#038; 1 &#038; 2 \\\\<br \/>\n0 &#038; 0 &#038; 2 &#038; 1 \\\\<br \/>\n3 &#038; 4 &#038; 0 &#038; 0 \\\\<br \/>\n4 &#038; 3 &#038; 0 &#038; 0 \\\\<br \/>\n\\end{bmatrix}\\sim \\begin{bmatrix}<br \/>\n0 &#038; 0 &#038; 3 &#038; 4 \\\\<br \/>\n0 &#038; 0 &#038; 4 &#038; 3 \\\\<br \/>\n1 &#038; 2 &#038; 0 &#038; 0 \\\\<br \/>\n2 &#038; 1 &#038; 0 &#038; 0<br \/>\n\\end{bmatrix}\\]\n<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv43() {\n  var htmlShow43 = document.getElementById(\"html-show43\");\n  if (htmlShow43.style.display === \"none\") {\n    htmlShow43.style.display = \"block\";\n  } else {\n    htmlShow43.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv43()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show43\" style=\"display: none;\">\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)\t<\/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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/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\">0<\/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\">3<\/span><span class=\"code_endofline\">,<\/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_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\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addcol<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}0 &amp; 0 &amp; 1 &amp; 2 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 2 &amp; 1 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\\\3 &amp; 4 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\4 &amp; 3 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}3 &amp; 4 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0\\\\4 &amp; 3 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1\\\\0 &amp; 0 &amp; 1 &amp; 2 &amp; 1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 2 &amp; 1 &amp; 0 &amp; 1 &amp; 0 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">EL<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">submatrix<\/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_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}0 &amp; 0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1\\\\1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">addrow<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">submatrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/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 class=\"code_function\">ident<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">4<\/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}3 &amp; 4 &amp; 0 &amp; 0\\\\4 &amp; 3 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 2\\\\0 &amp; 0 &amp; 2 &amp; 1\\\\1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0\\\\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\">(%i8)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">columnswap<\/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\">3<\/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\">columnswap<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}0 &amp; 0 &amp; 3 &amp; 4\\\\0 &amp; 0 &amp; 4 &amp; 3\\\\1 &amp; 2 &amp; 0 &amp; 0\\\\2 &amp; 1 &amp; 0 &amp; 0\\\\0 &amp; 0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1\\\\1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">ER<\/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_variable\">X<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}0 &amp; 0 &amp; 1 &amp; 0\\\\0 &amp; 0 &amp; 0 &amp; 1\\\\1 &amp; 0 &amp; 0 &amp; 0\\\\0 &amp; 1 &amp; 0 &amp; 0\\end{bmatrix}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">EL<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">A<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">ER<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}0 &amp; 0 &amp; 3 &amp; 4\\\\0 &amp; 0 &amp; 4 &amp; 3\\\\1 &amp; 2 &amp; 0 &amp; 0\\\\2 &amp; 1 &amp; 0 &amp; 0\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<ul>\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<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada \\(\\begin{bmatrix}2 &#038; \\operatorname{-}4 &#038; 3\\\\ 6 &#038; \\operatorname{-}8 &#038; 5\\\\ 6 &#038; 1 &#038; 7\\end{bmatrix}\\). Encontrar una matriz triangular superior que sea semejante por operaciones elementales.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv63() {\n  var htmlShow63 = document.getElementById(\"html-show63\");\n  if (htmlShow63.style.display === \"none\") {\n    htmlShow63.style.display = \"block\";\n  } else {\n    htmlShow63.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv63()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show63\" style=\"display: none;\">\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)\t<\/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_operator\">\u2212<\/span><span class=\"code_number\">4<\/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\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">8<\/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\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/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_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>\\[\\begin{bmatrix}2 &amp; -4 &amp; 3\\\\6 &amp; -8 &amp; 5\\\\6 &amp; 1 &amp; 7\\end{bmatrix}\\]<\/p>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 2 &amp; -4 &amp; 3\\\\0 &amp; 1 &amp; 0 &amp; 6 &amp; -8 &amp; 5\\\\0 &amp; 0 &amp; 1 &amp; 6 &amp; 1 &amp; 7\\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\">(%i3)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 2 &amp; -4 &amp; 3\\\\-3 &amp; 1 &amp; 0 &amp; 0 &amp; 4 &amp; -4\\\\0 &amp; 0 &amp; 1 &amp; 6 &amp; 1 &amp; 7\\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)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/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><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 2 &amp; -4 &amp; 3\\\\-3 &amp; 1 &amp; 0 &amp; 0 &amp; 4 &amp; -4\\\\-3 &amp; 0 &amp; 1 &amp; 0 &amp; 13 &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\">(%i5)\t<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">X<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">X<\/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\">13<\/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>\\[\\begin{bmatrix}1 &amp; 0 &amp; 0 &amp; 2 &amp; -4 &amp; 3\\\\-3 &amp; 1 &amp; 0 &amp; 0 &amp; 4 &amp; -4\\\\\\frac{27}{4} &amp; -\\frac{13}{4} &amp; 1 &amp; 0 &amp; 0 &amp; 11\\end{bmatrix}\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz propuesta en el video, realizar las mismas operaciones con maxima para concluir con la matriz escalonada que proporciona.<br \/>\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Matrices Semejantes por Transformaciones Elementales. Ej.3 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/ut2WSWWbAJ8?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<\/p>\n<\/blockquote>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Realizar como anteriormente con la matriz dada en:<br \/>\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Matrices semejantes por transformaciones elementales Ej.1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/V3-ghD34poQ?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<\/p>\n<\/blockquote>\n<hr \/>\n<p>A veces, podemos necesitar comandos que nos ayuden a simplificar expresiones, algunos son:<\/p>\n<ul>\n<li><strong>expand<\/strong>(<em>expr<\/em>): Expande la expresi\u00f3n <em>expr<\/em>. Los productos de sumas y de sumas con exponentes se multiplican, los numeradores de las expresiones racionales que son sumas se separan en sus respectivos t\u00e9rminos, y las multiplicaciones (tanto las que son conmutativas como las que no) se distribuyen sobre las sumas en todos los niveles de <em>expr<\/em>.<\/li>\n<li><strong>radcan<\/strong>(<em>expr<\/em>): Simplifica la expresi\u00f3n <em>expr<\/em>, que puede contener logaritmos, exponenciales y radicales, convirti\u00e9ndola a una forma can\u00f3nica, lo que significa que todas las expresiones funcionalmente equivalentes se reducen a una forma \u00fanica.<\/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<li><strong>fullratsimp<\/strong>(<em>expr<\/em>): Aplica repetidamente <em>ratsimp<\/em> a una expresi\u00f3n, seguida de simplificaciones no racionales, hasta que no se obtienen m\u00e1s transformaciones; entonces devuelve el resultado. <\/li>\n<\/ul>\n<p>Pod\u00e9is ver m\u00e1s en <a href=\"https:\/\/maxima.sourceforge.io\/docs\/manual\/es\/maxima_47.html\" rel=\"noopener\" target=\"_blank\">Funciones y variables para simplificaci\u00f3n<\/a><\/p>\n<h2>Rango de una matriz<\/h2>\n<p>Tenemos una funci\u00f3n que nos permite calcular el rango de una matriz:<\/p>\n<ul>\n<li><strong>rank<\/strong>(<em>M<\/em>): Calcula el rango de la matriz <em>M<\/em>.<\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Verifica el resultado del ejercicio<br \/>\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Rango de una matriz. Ej. 1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/iVSfMe8G8uA?start=9&#038;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<\/p>\n<\/blockquote>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Sea A:[[1,2,-3], [-2,0,4], [0,4,-2], [-2,-4,<strong>a<\/strong>]]. \u00bfCu\u00e1l es el valor de <strong>a<\/strong> para que el rango de la matriz sea par?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv63f2() {\n  var htmlShow63f2 = document.getElementById(\"html-show63f2\");\n  if (htmlShow63f2.style.display === \"none\") {\n    htmlShow63f2.style.display = \"block\";\n  } else {\n    htmlShow63f2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv63f2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show63f2\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Rango de una matriz. Ej.5 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/BY_dkD06MYY?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> Sea A:[[1,<strong>a<\/strong>,<strong>b<\/strong>,0], [2,2<strong>a<\/strong>,<strong>b<\/strong>,1], [2,3,<strong>b<\/strong>,0], [0,1,0,1]], con <strong>a<\/strong>,<strong>b<\/strong>\\(\\in\\mathbb{R}\\). \u00bfCu\u00e1l es el rango de la matriz seg\u00fan los par\u00e1metros dados?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv63f21() {\n  var htmlShow63f21 = document.getElementById(\"html-show63f21\");\n  if (htmlShow63f21.style.display === \"none\") {\n    htmlShow63f21.style.display = \"block\";\n  } else {\n    htmlShow63f21.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv63f21()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show63f21\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Rango de una matriz. Ej. 6 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/AAX_Z9JB1ok?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<h2>Inversa de una matriz<\/h2>\n<p>Recordad que para calcular la inversa de una matriz utilizaremos:<\/p>\n<blockquote><p>Sea \\(A\\) matriz cuadrada de orden \\(n\\). Si conseguimos mediante semejanza por transformaciones elementales una matriz tal que \\[[A\\, |\\, I_n] \\sim [I_n\\, |\\, B],\\]entonces \\(B\\) es la inversa de \\(A\\).\n<\/p><\/blockquote>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz A=[[1,1,0,0], [-1,1,-1,0], [0,1,1,1], [0,0,1,1]], \u00bfcu\u00e1nto es \\(\\mathbf{tr}(A^{-1})\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv63f2q() {\n  var htmlShow63f2q = document.getElementById(\"html-show63f2q\");\n  if (htmlShow63f2q.style.display === \"none\") {\n    htmlShow63f2q.style.display = \"block\";\n  } else {\n    htmlShow63f2q.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv63f2q()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show63f2q\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Matriz inversa con Maxima. Ej.10 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/Ei8BLnxM2k0?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<h3>Inversa mediante los comandos de maxima<\/h3>\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> Utiliza estos comandos para encontrar la matriz \\(X\\) tal que \\[\\begin{bmatrix}1&#038;2\\\\ 3&#038; 4\\end{bmatrix}X=\\begin{bmatrix}6&#038;3\\\\ 19&#038; 2\\end{bmatrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv63f() {\n  var htmlShow63f = document.getElementById(\"html-show63f\");\n  if (htmlShow63f.style.display === \"none\") {\n    htmlShow63f.style.display = \"block\";\n  } else {\n    htmlShow63f.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv63f()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show63f\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Matriz inversa con Maxima. Ej.9 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/guXfoesjuSI?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> Sea A=[[-1,1,3], [1,-2,0], [1,-2,1], [1,0,1]], y \\(L\\) su pseudoinversa. \u00bfcu\u00e1nto es el valor de \\(\\sum L[i,j]\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv63fw() {\n  var htmlShow63fw = document.getElementById(\"html-show63fw\");\n  if (htmlShow63fw.style.display === \"none\") {\n    htmlShow63fw.style.display = \"block\";\n  } else {\n    htmlShow63fw.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv63fw()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show63fw\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Pseudoinversa de una matriz. Ej.4 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/C0XMmKk7Izc?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<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 style=\"border: 0px;\">\n<td width=\"100%\"><strong>Ejercicio:<\/strong> Sea \\(A=\\begin{bmatrix}1&#038;1&#038;-1\\\\ 2&#038;0&#038;1\\end{bmatrix}\\), \u00bfcu\u00e1l es la suma de todos los elementos de su pseudoinversa? <\/td>\n<\/tr>\n<tr style=\"border: 0px;\">\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>3\/7<\/li>\n<li>5\/7<\/li>\n<li>12<\/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><br \/>\n<button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>B.)<\/strong><\/p>\n<p>Observa que la pseudoinversa es \\[\\begin{bmatrix}\\frac{3}{14} &#038; \\frac{5}{14}\\\\<br \/>\n\\frac{5}{14} &#038; \\mathop{-}\\left( \\frac{1}{14}\\right) \\\\<br \/>\n\\mathop{-}\\left( \\frac{3}{7}\\right)  &#038; \\frac{2}{7}\\end{bmatrix}\\]\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Matrices Si queremos utilizar matrices nos bastar\u00e1 con definirla mediante matrix(). Las filas de definimos como vectores: Ejercicio: Definir las matrices \\(A=\\begin{bmatrix}1 &#038; 2 &#038; 3\\\\ 4 &#038; 5 &#038; 6\\end{bmatrix}\\) y&#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-167","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\/167","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=167"}],"version-history":[{"count":9,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/167\/revisions"}],"predecessor-version":[{"id":214,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/167\/revisions\/214"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=167"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=167"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}