El siguiente ejercicio se ha propuesto en el examen de EvAU de Matem\u00e1ticas II (convocatoria de junio de 2023).<\/p>\n\n\n\n
Sean las matrices $X = \\left({\\begin{array}{c}a&b\\\\ c&0 \\end{array}} \\right)$, con $a,\\,b\\in\\mathbb{R}$, $A = \\left( {\\begin{array}{c} 2&1\\\\ 4&2 \\end{array}} \\right)$, $B = \\left( {\\begin{array}{c} 1&0\\\\ 2&0 \\end{array}} \\right)$.<\/p>\n\n\n\n
a) <\/strong>Determina las condiciones que tienen que cumplir los valores $a$, $b$, $c$ para que $A\\cdot X=B$.<\/p>\n\n\n\n b)<\/strong> Si adem\u00e1s queremos que $X$ sea sim\u00e9trica, \u00bfqu\u00e9 se debe cumplir? \u00bfC\u00f3mo es la matriz $X$ resultante?<\/p>\n\n\n\n a)<\/strong> $A\\cdot X=\\left( {\\begin{array}{c} 2&1\\\\ 4&2 \\end{array}} \\right)\\cdot\\left( {\\begin{array}{c} a&b\\\\ c&0 \\end{array}} \\right)=\\left( {\\begin{array}{c} 2a+c&2b\\\\ 4a+2c&4b \\end{array}} \\right)$. Entonces, para que $A\\cdot X=B$, se ha de cumplir que<\/p>\n\n\n\n $$\\left( {\\begin{array}{c} 2a+c&2b\\\\ 4a+2c&4b \\end{array}} \\right)=\\left( {\\begin{array}{c} 1&0\\\\ 2&0 \\end{array}} \\right)$$<\/p>\n\n\n\n De aqu\u00ed se deduce, por un lado, que $2a+c=1$ y $4a+2c=2$, que son ecuaciones equivalentes, ya que la segunda es el doble de la primera. Despejando $c$ de la primera ecuaci\u00f3n, tenemos que $c=1-2a$. Por otro lado, se tiene que $2b=0$ y que $4b=0$. O sea, que $b=0$. Por tanto, las condiciones que tienen que cumplir los valores $a$, $b$, $c$ para que $A\\cdot X=B$ son $c=1-2a,\\ b=0$. De este modo, para que $A\\cdot X=B$, debe ser<\/p>\n\n\n\n $$X=\\left( {\\begin{array}{c} a&0\\\\ 1-2a&0 \\end{array}} \\right)$$<\/p>\n\n\n\n b)<\/strong> $X=X^t \\Leftrightarrow \\left( {\\begin{array}{c} a&0\\\\ 1-2a&0 \\end{array}} \\right)=\\left( {\\begin{array}{c} a&1-2a\\\\ 0&0 \\end{array}} \\right) \\Leftrightarrow 1-2a=0 \\Leftrightarrow a=\\dfrac{1}{2}$.<\/p>\n\n\n\n Por tanto, la matriz resultante ser\u00e1 de la forma <\/p>\n\n\n\n $$X=\\left( {\\begin{array}{c} \\dfrac{1}{2}&0\\\\ 0&0 \\end{array}} \\right)$$<\/p>\n","protected":false},"excerpt":{"rendered":" El siguiente ejercicio se ha propuesto en el examen de EvAU de Matem\u00e1ticas II (convocatoria de junio de 2023). Enunciado Sean las matrices $X = \\left({\\begin{array}{c}a&b\\\\ c&0 \\end{array}} \\right)$, con $a,\\,b\\in\\mathbb{R}$, $A = \\left( {\\begin{array}{c} 2&1\\\\ 4&2 \\end{array}} \\right)$, $B = \\left( {\\begin{array}{c} 1&0\\\\ 2&0 \\end{array}} \\right)$. a) Determina las condiciones que tienen que cumplir […]<\/p>\n","protected":false},"author":1,"featured_media":909,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25,4,23,6],"tags":[26,30,32,27,29,28,31],"class_list":{"0":"post-888","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","6":"hentry","7":"category-algebra-bachillerato","8":"category-bachillerato","9":"category-evau","10":"category-posts","11":"tag-evau","12":"tag-evau-matematicas","13":"tag-evau-matematicas-resueltos","14":"tag-evau-uclm","15":"tag-evau-uclm-matematicas","16":"tag-evau-uclm-resueltos","17":"tag-uclm-matematicas","19":"post-with-thumbnail","20":"post-with-thumbnail-large"},"_links":{"self":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/888","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=888"}],"version-history":[{"count":20,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/888\/revisions"}],"predecessor-version":[{"id":908,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/posts\/888\/revisions\/908"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=\/wp\/v2\/media\/909"}],"wp:attachment":[{"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=888"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=888"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/matematicastro.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=888"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Soluci\u00f3n<\/h4>\n\n\n\n