{"id":117746,"date":"2020-01-29T18:24:34","date_gmt":"2020-01-29T17:24:34","guid":{"rendered":"https:\/\/www.superprof.es\/diccionario\/algebralineal\/teorema-rouche.html"},"modified":"2026-10-07T17:43:28","modified_gmt":"2026-10-07T15:43:28","slug":"teorema-rouche","status":"publish","type":"post","link":"https:\/\/www.superprof.es\/diccionario\/matematicas\/algebralineal\/teorema-rouche.html","title":{"rendered":"Teorema de rouch\u00e9-fr\u00f6benius"},"content":{"rendered":"<section id=\"sec\" style=\"\">\r\n          \n\n<p class=\"wp-block-paragraph\">\r\n\t\t\t\t\t\t\t\tLa condici\u00f3n necesaria y suficiente para que un sistema de m ecuaciones y n inc\u00f3gnitas tenga soluci\u00f3n es que el <strong>rango de la matriz de los coeficientes y el de la matriz ampliada sean iguales.\t\t\t\t\t\t\t<\/strong><\/p>\n\n\r\n\t\t\t\t\t\t\t\r\n\t\t\t\t\t\t\t\r\n\t\t\t\t\t\t\t\r\n\t\t\t  \n\n<ul class=\"wp-block-list\">\r\n\t\t\t\t\t\t\t\t<li><span ><strong>r = r'<\/strong><\/span>               <strong>Sistema Compatible.<\/strong> \r\n\t\t\t\t\t\t\t\r\n\t\t\t\t\t\t          <ul>\r\n\t\t\t\t\t\t\t\t\r\n\t\t\t\t\t\t\t\t\t<li>\r\n\t\t\t\t\t\t\r\n\t\t\t\t\t\t\t\t\t\t<span><strong>r = r'= n<\/strong><\/span>   <strong>Sistema Compatible Determinado.<\/strong>\t\t\t\t\t\t\t\t  <\/li>\r\n<li>\r\n\t\t\t\t\t\t\t\t\t\t\r\n\t\t\t\t\t\t\t\t\t\t<strong><span>r = r'\u2260 n<\/span><\/strong>   <strong>Sistema Compatible Indeterminado.<\/strong>\t\t\t\t\t\t\t\t\t<\/li>\r\n\t\t\t\t\t\t\t\t\t\r\n\t\t\t\t\t\t\t\t<\/ul>\r\n\t\t\t\t\t\t\t\t\r\n\t\t\t\t\t\t\t\t<\/li>\r\n\t\t\t\t\t\t\t  <li><span><strong>r \u2260 r'<\/strong><\/span>               <strong>Sistema Incompatible.<\/strong>\r\n\t\t\t\t\t\t\t\t<\/li>\r\n\t\t\t\t\t\t\t<\/ul>\n\n\r\n                             \r\n\r\n\t\t\t  \n\n<h2 class=\"wp-block-heading\">Ejemplo<\/h2>\n\n\r\n\t\t\t\t\t\t\t\n\n<p class=\"wp-block-paragraph\">\r\n                                <img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-1.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 1\"  >\t\t\t\t\t\t\t\t\r\n                            <\/p>\n\n\r\n                            \n\n<p class=\"wp-block-paragraph\">\r\n                            \n\n<h2 class=\"wp-block-heading\">1.<\/h2>\n\n Tomamos la matriz de los coeficientes y le hallamos el rango.\t\t\t\t\t\t  \r\n                            <\/p>\n\n\r\n                                            \r\n                            \n\n<p class=\"wp-block-paragraph\">\r\n                                <img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-2.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 2\"  >\r\n                            <\/p>\n\n \r\n                            \n\n<p class=\"wp-block-paragraph\">\r\n                                        <img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-3.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 3\"  >\r\n                            <\/p>\n\n\r\n                            \n\n<p class=\"wp-block-paragraph\"><span class=\"sol\">r(A) = 3<\/span><\/p>\n\n\r\n                            \n\n<p class=\"wp-block-paragraph\">\n\n<h2 class=\"wp-block-heading\">2. <\/h2>\n\n Hallamos el rango de la matriz ampliada<\/p>\n\n\r\n                            \n\n<p class=\"wp-block-paragraph\"><img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-4.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 4\"  ><\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\">  <img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-5.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 5\"   ><\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\"><span class=\"sol\">r(A') = 3<\/span><\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\">\n\n<h2 class=\"wp-block-heading\">3. <\/h2>\n\n Aplicamos el <strong>teorema de Rouch\u00e9. <\/strong><\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\"><img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-6.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 6\"  ><\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\"><img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-7.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 7\"  ><\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\">\n\n<h2 class=\"wp-block-heading\">4.<\/h2>\n\n Se resuelve el sistema, si \u00e9ste no es incompatible, por la regla de Cramer o por el m\u00e9todo de Gauss<\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\">Tomamos el sistema que corresponde a la submatriz de orden 3, que tiene rango 3, y lo resolvemos.<\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\"> <img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-8.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 8\"  ><\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\"> <img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-9.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 9\"  ><\/p>\n\n\r\n                             \n\n<p class=\"wp-block-paragraph\"> <img decoding=\"async\" src=\"https:\/\/www.superprof.es\/diccionario\/wp-content\/uploads\/2020\/01\/teorema-de-rouch&eacute;-fr&ouml;benius-10.gif\" alt=\"Explicaciones y ejemplos de teorema de rouch&eacute;-fr&ouml;benius - 10\"  ><\/p>\n\n\n<\/section>","protected":false},"excerpt":{"rendered":"<p>La condici\u00f3n necesaria y suficiente para que un sistema de m ecuaciones y n inc\u00f3gnitas tenga soluci\u00f3n es que el rango de la matriz de los coeficientes y el de la matriz ampliada sean iguales. r = r&#8217; Sistema Compatible. r = r&#8217;= n Sistema Compatible Determinado. r = r&#8217;\u2260 n Sistema Compatible Indeterminado. r [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_seopress_robots_primary_cat":"","_seopress_titles_title":"","_seopress_titles_desc":"","_seopress_robots_index":"","advgb_blocks_editor_width":"","advgb_blocks_columns_visual_guide":"","footnotes":""},"categories":[372],"tags":[364],"class_list":["post-117746","post","type-post","status-publish","format-standard","hentry","category-algebralineal","tag-t"],"acf":[],"author_meta":{"display_name":"Andra","author_link":"https:\/\/www.superprof.es\/diccionario\/author\/andra"},"featured_img":null,"coauthors":[],"tax_additional":{"categories":{"linked":["<a href=\"https:\/\/www.superprof.es\/diccionario\/matematicas\/algebralineal\" class=\"advgb-post-tax-term\">\u00c1lgebra lineal<\/a>"],"unlinked":["<span class=\"advgb-post-tax-term\">\u00c1lgebra lineal<\/span>"]},"tags":{"linked":["<a href=\"https:\/\/www.superprof.es\/diccionario\/matematicas\/algebralineal\" class=\"advgb-post-tax-term\">t<\/a>"],"unlinked":["<span class=\"advgb-post-tax-term\">t<\/span>"]}},"comment_count":"0","relative_dates":{"created":"Publicado 7 a\u00f1os hace","modified":"Actualizado 1 mes hace"},"absolute_dates":{"created":"Publicado el 29 enero 2020","modified":"Actualizado el 7 octubre 2026"},"absolute_dates_time":{"created":"Publicado el 29 enero 2020 18 h 24 min","modified":"Actualizado el 7 octubre 2026 17 h 43 min"},"featured_img_caption":"","series_order":"","_links":{"self":[{"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/posts\/117746","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/comments?post=117746"}],"version-history":[{"count":5,"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/posts\/117746\/revisions"}],"predecessor-version":[{"id":140963,"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/posts\/117746\/revisions\/140963"}],"wp:attachment":[{"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/media?parent=117746"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/categories?post=117746"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.superprof.es\/diccionario\/wp-json\/wp\/v2\/tags?post=117746"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}