{"id":142,"date":"2025-10-01T11:45:09","date_gmt":"2025-10-01T09:45:09","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=142"},"modified":"2025-09-25T12:46:07","modified_gmt":"2025-09-25T10:46:07","slug":"mathbio-inversa-de-una-matriz-y-determinantes","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=142","title":{"rendered":"MathBio: Inversa de una matriz y Determinantes."},"content":{"rendered":"<h2>Inversa de una matriz<\/h2>\n<p>Definimos la inversa de una matriz cuadrada \\(A=[a_{ij}]\\in \\mathcal{M}_{n}(\\mathbb{R})\\) como la matriz \\(B=[b_{ij}]\\in \\mathcal{M}_{n}(\\mathbb{R})\\) tal que \\[AB=BA=I_n.\\]<\/p>\n<p>El procedimiento que damos para calcular la inversa, es el de realizar operaciones elementales entre filas o columnas, que conoc\u00e9is como m\u00e9todo de Gauss. Ser\u00eda el siguiente: Sea \\(A\\) la matriz, y consideremos la matriz formada por \\([A\\, |\\, I_n]\\). Si conseguimos mediante semejanza por transformaciones elementales una matriz tal que<\/p>\n<p>\\[[A\\, |\\, I_n] \\sim [I_n\\, |\\, B],\\]<\/p>\n<p>entonces \\(B\\) es la inversa de \\(A\\).<\/p>\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 la traza de su inversa? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4f() {\n  var htmlShow4f = document.getElementById(\"html-show4f\");\n  if (htmlShow4f.style.display === \"none\") {\n    htmlShow4f.style.display = \"block\";\n  } else {\n    htmlShow4f.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4f()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4f\" style=\"display: none;\">\n\\[\\begin{pmatrix}1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n-1 &#038; 1 &#038; -1 &#038; 0 &#038; 0 &#038; 1 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 1 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 1\\end{pmatrix}\\overset{f_2+f_1}{\\sim }\\begin{pmatrix}1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 2 &#038; -1 &#038; 0 &#038; 1 &#038; 1 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 1 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 1\\end{pmatrix}\\]<\/p>\n<p>\\[\\overset{f_3\\leftrightarrow f_2}{\\sim }\\begin{pmatrix}1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 1 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0\\\\<br \/>\n0 &#038; 2 &#038; -1 &#038; 0 &#038; 1 &#038; 1 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 1\\end{pmatrix}\\overset{f_3-2 f_2}{\\sim }\\begin{pmatrix}1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 1 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; -3 &#038; -2 &#038; 1 &#038; 1 &#038; -2 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 1\\end{pmatrix}\\]<\/p>\n<p>\\[\\overset{f_4\\leftrightarrow f_3}{\\sim }\\begin{pmatrix}1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 1 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 1\\\\<br \/>\n0 &#038; 0 &#038; -3 &#038; -2 &#038; 1 &#038; 1 &#038; -2 &#038; 0\\end{pmatrix}\\overset{f_4+3 f_3}{\\sim }\\begin{pmatrix}1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 1 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 1\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1 &#038; 1 &#038; 1 &#038; -2 &#038; 3\\end{pmatrix}\\]<br \/>\n\\[\\overset{\\begin{array}{l}<br \/>\nf_3-f_4 \\\\<br \/>\nf_2-f_4 \\\\<br \/>\nf_2-f_3 \\\\<br \/>\nf_1-f_2<br \/>\n\\end{array}}{\\sim }\\begin{pmatrix}1 &#038; 0 &#038; 0 &#038; 0 &#038; 1 &#038; 0 &#038; -1 &#038; 1\\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; 0 &#038; 0 &#038; 0 &#038; 1 &#038; -1\\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; 0 &#038; -1 &#038; -1 &#038; 2 &#038; -2\\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1 &#038; 1 &#038; 1 &#038; -2 &#038; 3\\end{pmatrix}\\]\n<\/p><\/div>\n<hr \/>\n<blockquote>\n<p><strong>Proposici\u00f3n:<\/strong> Dadas las matrices cuadradas regulares del mismo orden \\(A\\) y \\(B\\)<\/p>\n<ol>\n<li>\\((A^t)^{-1}=(A^{-1})^t\\)<\/li>\n<li>\\((A\\,B)^{-1}=B^{-1}\\cdot A^{-1}\\)<\/li>\n<\/ol>\n<\/blockquote>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Para todo \\(p\\in\\mathbb{Z}^+\\) y  \\(A\\in\\mathcal{M}_n(\\mathbb{R})\\) un matriz regular; entonces, la inversa de \\(A^p\\) es \\(\\left(A^{-1}\\right)^p\\). \u00bfVerdadero o falso? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv5w() {\n  var htmlShow5w = document.getElementById(\"html-show5w\");\n  if (htmlShow5w.style.display === \"none\") {\n    htmlShow5w.style.display = \"block\";\n  } else {\n    htmlShow5w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv5w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show5w\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal -  Propiedades de las matrices regulares -Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/v8zybNqJ7dA?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>Determinante de una matriz<\/h2>\n<p>Para que sea m\u00e1s f\u00e1cil definimos los determinantes de forma recursiva, utilizando el valor de un determinante de orden dos y la <a href=\"http:\/\/es.wikipedia.org\/wiki\/Teorema_de_Laplace\" target=\"_blank\" rel=\"noopener noreferrer\">Regla de Laplace<\/a>:<\/p>\n<blockquote>\n<ul>\n<li>Sea \\(A=\\begin{bmatrix}a_{11}&#038;a_{12}\\\\ a_{21}&#038;a_{22}\\end{bmatrix}\\in\\mathcal{M}_2(\\mathbb{R})\\), definimos el determinante de \\(A\\), como \\[|A|=a_{11}a_{22}-a_{12}a_{21}.\\]<\/li>\n<li> Para todo \\(n>2\\) y \\(A\\in\\mathcal{M}_n(\\mathbb{R})\\), definimos \\[|A|=\\sum _{j=1}^{n}a_{1j}\\;A_{1j},\\] donde \\(A_{1j}=(-1)^{(1+j)}\\;\\alpha _{1j}\\), siendo \\(\\alpha _{1j}\\) el determinante de orden \\(n-1\\) que queda tras eliminar de la matriz \\(A\\) la fila 1 y la columna \\(j\\).\n<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\[A=\\begin{bmatrix}<br \/>\n-1 &#038; 2 &#038; 3 &#038; 1\\\\<br \/>\n1 &#038; 0 &#038; 5 &#038; 0 \\\\<br \/>\n2 &#038; -1 &#038; 1 &#038; 2 \\\\<br \/>\n3 &#038; 4 &#038; 7 &#038; 0<br \/>\n\\end{bmatrix}\\in\\mathcal{M}_4(\\mathbb{R}),\\]  \u00bfcu\u00e1nto es su determinante? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4aq() {\n  var htmlShow4aq = document.getElementById(\"html-show4aq\");\n  if (htmlShow4aq.style.display === \"none\") {\n    htmlShow4aq.style.display = \"block\";\n  } else {\n    htmlShow4aq.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4aq()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4aq\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Determinante. Definici\u00f3n por recursi\u00f3n - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/Lf6CoElqoRg?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\[A=\\begin{bmatrix}<br \/>\n1 &#038; 2 &#038; 0 &#038; 0\\\\<br \/>\n-1 &#038; 1 &#038; -1 &#038; 0 \\\\<br \/>\n0 &#038; 2 &#038; -1 &#038; 2 \\\\<br \/>\n0 &#038; 0 &#038; 2 &#038; 1<br \/>\n\\end{bmatrix}\\in\\mathcal{M}_4(\\mathbb{R}),\\]  \u00bfcu\u00e1nto es su determinante? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4() {\n  var htmlShow4 = document.getElementById(\"html-show4\");\n  if (htmlShow4.style.display === \"none\") {\n    htmlShow4.style.display = \"block\";\n  } else {\n    htmlShow4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4\" style=\"display: none;\">\nAplicando la definici\u00f3n:<br \/>\n\\[\\begin{vmatrix}<br \/>\n1 &#038; 2 &#038; 0 &#038; 0\\\\<br \/>\n-1 &#038; 1 &#038; -1 &#038; 0 \\\\<br \/>\n0 &#038; 2 &#038; -1 &#038; 2 \\\\<br \/>\n0 &#038; 0 &#038; 2 &#038; 1<br \/>\n\\end{vmatrix}=\\begin{vmatrix}<br \/>\n1 &#038; -1 &#038; 0 \\\\<br \/>\n2 &#038; -1 &#038; 2 \\\\<br \/>\n0 &#038; 2 &#038; 1<br \/>\n\\end{vmatrix}-2<br \/>\n\\begin{vmatrix}<br \/>\n-1 &#038; -1 &#038; 0 \\\\<br \/>\n0 &#038; -1 &#038; 2 \\\\<br \/>\n0 &#038; 2 &#038; 1<br \/>\n\\end{vmatrix}\\]\n<\/div>\n<hr \/>\n<p>La definici\u00f3n cl\u00e1sica y su significado puede verse en <a href=\"http:\/\/en.wikipedia.org\/wiki\/Determinant\" target=\"_blank\" rel=\"noopener noreferrer\">Determinante<\/a>. En este enlace pod\u00e9is encontrar tambi\u00e9n propiedades importantes. Recordad estas propiedades porque ser\u00e1n muy importantes para aprender bien este tema.<\/p>\n<blockquote>\n<p><strong>Regla de Laplace:<\/strong> El determinante de una matriz es independiente de la fila o columna que elijamos en el paso 2 anterior.<br \/>\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Determinantes. Regla de Laplace - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/a3A3IAynQaI?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<h3>Propiedades de los determinantes<\/h3>\n<p>Asumamos \\(A\\) y \\(B\\) dos matrices cuadradas del mismo orden,<\/p>\n<ol>\n<li>\\(|A|=|A^t|\\)<\/li>\n<li>Si \\(B\\) es el resultado de hacer una transformaci\u00f3n elemental por fila(columna) a la matriz \\(A\\), \\(A\\overset{f_i+\\lambda f_j\\\\ (c_i+\\lambda c_j)}{\\sim}B\\Rightarrow|A|=|B|\\)<\/li>\n<li>Si \\(B\\) es el resultado de intercambiar una fila(columna) de la matriz \\(A\\), \\(A\\overset{f_i \\leftrightarrow f_j\\\\ (c_i\\leftrightarrow c_j)}{\\sim}B\\Rightarrow|A|=-|B|\\)<\/li>\n<li>Si \\(B\\) es el resultado de multiplicar una fila(columna) de la matriz \\(A\\) por un escalar, \\(A\\overset{f_i = \\lambda f_i\\\\ (c_i=\\lambda c_i)}{\\sim}B\\Rightarrow|B|=\\lambda |A|\\)<\/li>\n<li>\\(\\begin{vmatrix}a_{11}&amp;a_{12}\\\\ a+b &amp; c+d\\end{vmatrix}=\\begin{vmatrix}a_{11}&amp;a_{12}\\\\ a &amp; c\\end{vmatrix}+\\begin{vmatrix}a_{11}&amp;a_{12}\\\\ b &amp; d\\end{vmatrix}\\). De igual modo podemos hacerlo para toda matriz cuadrada de orden \\(n\\).<\/li>\n<li>\\(|A\\,B|=|A|\\cdot |B|\\)<\/li>\n<\/ol>\n<p>Consecuencia de las propiedades anteriores son estos resultados:<\/p>\n<blockquote>\n<ol>\n<li>El determinante de una matriz con una fila, o columna, todo ceros vale cero.<\/li>\n<li>El determinante de una matriz con dos filas, o columnas, proporcionales vale cero.<\/li>\n<li>El determinante de una matriz triangular es igual al producto de los elementos de la diagonal principal.<\/li>\n<\/ol>\n<\/blockquote>\n<blockquote>\n<p><strong>Proposici\u00f3n:<\/strong> Una matriz cuadrada no tiene inversa si su determiante  es cero.<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\(A\\)= [[1,-5,3],[-1,-3,4],[-3,7,x]], \u00bfcu\u00e1l es el valor de \\(x\\) para que la matriz no tenga inversa?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4f1() {\n  var htmlShow4f1 = document.getElementById(\"html-show4f1\");\n  if (htmlShow4f1.style.display === \"none\") {\n    htmlShow4f1.style.display = \"block\";\n  } else {\n    htmlShow4f1.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4f1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4f1\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Inversa de una matriz. Ej.8 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/JwTufPUzRHI?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> Cu\u00e1l es el valor del determinante  \\[\\begin{vmatrix}<br \/>\n1 &#038; 1 &#038; 1 &#038; 1\\\\<br \/>\n1 &#038; 2 &#038; 2 &#038; 2\\\\<br \/>\n1 &#038; 2 &#038; 3 &#038; 3\\\\<br \/>\n1 &#038; 2 &#038; 3 &#038; 4<br \/>\n\\end{vmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv7() {\n  var htmlShow7 = document.getElementById(\"html-show7\");\n  if (htmlShow7.style.display === \"none\") {\n    htmlShow7.style.display = \"block\";\n  } else {\n    htmlShow7.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv7()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show7\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Determinantes: EJ.3 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/iWFECtAih2U?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>Menor de una matriz<\/h2>\n<p>Un menor de una matriz \\(A\\) es el determinante de una submatriz cuadrada de \\(A\\). <\/p>\n<p> La definici\u00f3n de menor nos da pie a otro resultado muy interesante. Podemos extender la definici\u00f3n de menor para una matriz no cuadrada a cualquier determinante de una submatriz cuadrada. En este caso:\n<\/p>\n<blockquote>\n<p><strong> Teorema.<\/strong> Si \\(A\\) es una matriz, el rango de \\(A\\) es el orden del mayor menor de \\(A\\) no nulo.\n<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Cu\u00e1l es rango de la matriz  \\[\\begin{bmatrix}<br \/>\n1 &#038; -3 &#038; -1 &#038; -1\\\\<br \/>\n1 &#038; 5 &#038; 3 &#038; 3\\\\<br \/>\n1 &#038; 1 &#038; 1 &#038; 1\\\\<br \/>\n3 &#038; 7 &#038; 5 &#038; 5<br \/>\n\\end{bmatrix}\\] <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv16() {\n  var htmlShow16 = document.getElementById(\"html-show16\");\n  if (htmlShow16.style.display === \"none\") {\n    htmlShow16.style.display = \"block\";\n  } else {\n    htmlShow16.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv16()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show16\" style=\"display: none;\">\nSoluci\u00f3n: \\(2\\)\n<\/div>\n<hr \/>\n<h3>Bibliograf\u00eda<\/h3>\n<ul>\n<li>Cap\u00edtulo 4 de \u00c1lgebra lineal y sus aplicaciones. 5\u00ba edici\u00f3n, David C. Lay. Pearson. 2016.<\/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><br \/>\nDada la matriz \\(\\begin{bmatrix}1&#038;0&#038;0&#038;1\\\\0&#038;1&#038;2&#038;-1\\\\ 0&#038;1&#038;-1&#038;0\\\\ 0&#038;2&#038;0&#038;-1\\end{bmatrix}\\), \u00bfcu\u00e1l es el valor de su determinante?<\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>-1<\/li>\n<li>1<\/li>\n<li>2<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>B.)<\/strong><\/p>\n<p>\\[\\begin{vmatrix}1 &#038; 0 &#038; 0 &#038; 1\\\\<br \/>\n0 &#038; 1 &#038; 2 &#038; -1\\\\<br \/>\n0 &#038; 1 &#038; -1 &#038; 0\\\\<br \/>\n0 &#038; 2 &#038; 0 &#038; -1\\end{vmatrix}=\\begin{vmatrix}1 &#038; 2 &#038; -1\\\\<br \/>\n1 &#038; -1 &#038; 0\\\\<br \/>\n2 &#038; 0 &#038; -1\\end{vmatrix}\\overset{f_3-f_1}{=}\\begin{vmatrix}1 &#038; 2 &#038; -1\\\\<br \/>\n1 &#038; -1 &#038; 0\\\\<br \/>\n1 &#038; -2 &#038; 0\\end{vmatrix}=\\] \\[=-\\begin{vmatrix}1 &#038; -1 \\\\ 1 &#038; -2 \\end{vmatrix}\\overset{f_2-f_1}{=}-\\begin{vmatrix}1 &#038; -1 \\\\ 0 &#038; -1 \\end{vmatrix}=1\\]\n<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Inversa de una matriz Definimos la inversa de una matriz cuadrada \\(A=[a_{ij}]\\in \\mathcal{M}_{n}(\\mathbb{R})\\) como la matriz \\(B=[b_{ij}]\\in \\mathcal{M}_{n}(\\mathbb{R})\\) tal que \\[AB=BA=I_n.\\] El procedimiento que damos para calcular la inversa, es el de&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-142","post","type-post","status-publish","format-standard","hentry","category-mathbio"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/142","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=142"}],"version-history":[{"count":4,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/142\/revisions"}],"predecessor-version":[{"id":163,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/142\/revisions\/163"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=142"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=142"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=142"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}