{"id":299,"date":"2023-07-06T17:07:12","date_gmt":"2023-07-06T17:07:12","guid":{"rendered":"https:\/\/mathority.org\/it\/estensione-di-una-matrice-in-funzione-di-un-parametro-esempi-ed-esercizi-risolti-di-matrici-2x2-3x3-3x4-4x4\/"},"modified":"2023-07-06T17:07:12","modified_gmt":"2023-07-06T17:07:12","slug":"estensione-di-una-matrice-in-funzione-di-un-parametro-esempi-ed-esercizi-risolti-di-matrici-2x2-3x3-3x4-4x4","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/estensione-di-una-matrice-in-funzione-di-un-parametro-esempi-ed-esercizi-risolti-di-matrici-2x2-3x3-3x4-4x4\/","title":{"rendered":"Intervallo di un array basato su un parametro"},"content":{"rendered":"<p>In questa pagina vedrai come calcolare il <strong>rango di una tabella in base a un parametro.<\/strong> Troverai anche esempi passo passo ed esercizi risolti su come trovare l&#8217;intervallo di una matrice in base a un parametro.<\/p>\n<p> Per comprendere appieno la procedura per studiare il rango delle matrici con parametri, \u00e8 importante sapere gi\u00e0 <a href=\"https:\/\/mathority.org\/it\">come calcolare il rango di una matrice in base ai determinanti<\/a> . Quindi ti consigliamo di imparare queste due cose prima di continuare a leggere.<\/p>\n<h2 class=\"wp-block-heading\"> Come calcolare l&#8217;intervallo di un array in base a un parametro. Esempio:<\/h2>\n<ul>\n<li> Determina l&#8217;intervallo della matrice A in base a diversi valori dei parametri\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0aa5688f2845a0225149f448466c943c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\begin{pmatrix} a+1 &amp; -1 &amp; a+1 \\\\[1.1ex] 0 &amp; -1 &amp; 0   \\\\[1.1ex] 1 &amp; -2 &amp; a  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"198\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> La matrice A avr\u00e0 al massimo rango 3, perch\u00e9 \u00e8 una matrice di ordine 3. Pertanto, la prima cosa da fare \u00e8 <strong>risolvere il determinante dell&#8217;intera matrice 3&#215;3<\/strong> con <a href=\"https:\/\/mathority.org\/it\/determinanti-3x3-esempi-di-regole-sarrus-ed-esercizi-risolti\/\">la regola di Sarrus<\/a> , per vedere se pu\u00f2 essere di rango 3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-835a881061438326519f4660b4c394fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} a+1 &amp; -1 &amp; a+1 \\\\[1.1ex] 0 &amp; -1 &amp; 0   \\\\[1.1ex] 1 &amp; -2 &amp; a  \\end{vmatrix} &amp; =-a(a+1)+0+0+a+1-0-0 \\\\ &amp; =-a^2-a+a+1  \\\\[1.5ex] &amp; =-a^2+1 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"429\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Il risultato del determinante \u00e8 una funzione del parametro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> . <strong>Impostiamo quindi il risultato uguale a 0<\/strong> per vedere quando la tabella sar\u00e0 di rango 2 e quando sar\u00e0 di rango 3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7cf08fe725290ac099f54916fa4c5dcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -a^2+1 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"93\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> E risolviamo l&#8217;equazione risultante: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-18b6f04242243eeefa0cd5892b29f4d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a^2 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ad7c0d92bbec913193a85949c7a0bfa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\sqrt{a^2} = \\sqrt{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -1px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1191c881d84f673236382966b4e709ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{a = \\pm 1}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Pertanto, quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> che sia +1 o -1, il determinante 3\u00d73 sar\u00e0 0 e, quindi, il rango della matrice non sar\u00e0 3. Invece, quando<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e8 diverso da +1 e -1, il determinante sar\u00e0 diverso da 0 e, quindi, la matrice sar\u00e0 di rango 3.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-60e6f80d73c96b28458d7790d98d0a5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c}  \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq +1,-1 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"354\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ora vediamo cosa succede quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbdf9897658213f9f2ad0b6a3d8d87cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{a=+1} :\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"65\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-53dde6f61dc01cac5c0a0705c44a7433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = +1 \\longrightarrow A= \\begin{pmatrix} 2 &amp; -1 &amp; 2 \\\\[1.1ex] 0 &amp; -1 &amp; 0   \\\\[1.1ex] 1 &amp; -2 &amp; 1  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"230\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Come abbiamo visto in precedenza, quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e8 1 il determinante della matrice \u00e8 0. Non pu\u00f2 quindi essere di rango 3. Proviamo ora a calcolare un <a href=\"https:\/\/mathority.org\/it\/determinanti-2x2-esempi-ed-esercizi-risolti\/\">determinante 2\u00d72<\/a> diverso da 0 all&#8217;interno della matrice, ad esempio quello dell&#8217;angolo in alto a sinistra:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d291f322f9d3f392e46568817e531a84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle   \\begin{vmatrix} 2 &amp; -1 \\\\[1.1ex] 0 &amp; -1 \\end{vmatrix} =-2-0= -2 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Il determinante di ordine 2 \u00e8 diverso da 0. Pertanto, quando il parametro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> o +1, il <strong>rango della matrice sar\u00e0 2:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d00c47041db87183749744eaf6789fd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c}  \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = +1 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Una volta che vediamo l&#8217;intervallo della matrice quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-910ad8735da02f7dffe9cd0fda341d6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a \\neq +1,-1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"86\" style=\"vertical-align: -4px;\"><\/p>\n<p> e quando<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-10f3012b6955e51b81c57a6e2e57b7df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=+1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"55\" style=\"vertical-align: -2px;\"><\/p>\n<p> Vediamo cosa succede quando<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d04c75a36ec68cca9920060cc558b99_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{a = -1} :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f723d9c6b9f786b8c405ac7ec2d8bf1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = -1 \\longrightarrow A=  \\begin{pmatrix} 0 &amp; -1 &amp; 0 \\\\[1.1ex] 0 &amp; -1 &amp; 0   \\\\[1.1ex] 1 &amp; -2 &amp; -1  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Come abbiamo visto all&#8217;inizio, quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> es -1 e il determinante della matrice \u00e8 0. Pertanto, non pu\u00f2 essere impostato al rango 3. Pertanto, dovremmo cercare di incontrare nella matrice un determinante di 2\u00d72 diverso da 0, ad esempio il pi\u00f9 basso parte della matrice. SINISTRA:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdc9bd6d9ad083e1e38f53079aebb5e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle   \\begin{vmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; -2  \\end{vmatrix} = 0-(-1)= 1\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Il determinante della dimensione 2 \u00e8 diverso da 0. Pertanto, quando il parametro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> o -1, il <strong>rango della tabella sar\u00e0 2:<\/strong><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d12346bae2f327e7e1ee6c5276a599cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = -1 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Abbiamo quindi trovato 3 casi diversi in cui il rango della matrice A dipende dal valore che assume il parametro<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4ae924c776e55c0f2987a783307cd9fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> Ecco il <strong>riepilogo<\/strong> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc3a7ebea32c871ab7971a276decc60a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq +1,-1 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = +1 \\ \\longrightarrow \\ Rg(A)=2} \\\\[3ex]  \\color{black} \\bm{a = -1 \\ \\longrightarrow \\ Rg(A)=2} \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"354\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ora che sai come discutere la gamma di matrici dipendenti dai parametri, puoi esercitarti a svolgere gli esercizi passo passo riportati di seguito. Per risolverli, le <a href=\"https:\/\/mathority.org\/it\/proprieta-dei-determinanti-esempi-ed-esercizi-2x2-3x3\/\">propriet\u00e0 dei determinanti<\/a> ti aiuteranno sicuramente, quindi se non ti sono molto chiare su di esse, ti consiglio di dare prima un&#8217;occhiata alla pagina collegata, dove ognuno di essi \u00e8 spiegato con degli esempi.<\/p>\n<h2 class=\"wp-block-heading\"> Risolti i problemi relativi all&#8217;intervallo della matrice basata su parametri<\/h2>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Studiare l&#8217;intervallo della tabella seguente in base al valore del parametro <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7f53b08bcf2e2660dbb7c0aeb6fd369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; 1 &amp; a \\\\[1.1ex] 2 &amp; 2 &amp; -4 \\\\[1.1ex] 2 &amp; 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La matrice A avr\u00e0 al massimo rango 3, perch\u00e9 \u00e8 una matrice 3\u00d73. Pertanto, la prima cosa da fare \u00e8 risolvere il determinante dell&#8217;intera matrice (con la regola di Sarrus), per vedere se pu\u00f2 essere di rango 3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d2539698cbcf9f06d2890d17da76174f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; 1 &amp; a \\\\[1.1ex] 2 &amp; 2 &amp; -4 \\\\[1.1ex] 2 &amp; 1 &amp; 0 \\end{vmatrix}  =0-8+2a-4a+12-0 =-2a+4\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"382\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Impostiamo il risultato uguale a 0 per vedere quando l&#8217;array sar\u00e0 di rango 2 e quando di rango 3: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7042ae953fdcbe91d08fa963be26f7c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2a+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"94\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-90183d93145fd04e7a774c8a72bc3f1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2a=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"78\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-21d4999dede651fdb38c5b047b8e805d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a=\\cfrac{-4}{-2} = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"97\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pertanto, quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e8 diverso da 2, il determinante 3\u00d73 sar\u00e0 diverso da 0 e, quindi, il rango della matrice sar\u00e0 3.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a4c4e0bfd1194afe82d8807c033e7551_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq 2 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"310\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora vediamo cosa succede quando <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f2abbabd80372bf9bc248f12cebd5fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=2 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c131e19dd5d5c0d7826306103b4e118b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = 2 \\longrightarrow A= \\begin{pmatrix} 3 &amp; 1 &amp; 2 \\\\[1.1ex] 2 &amp; 2 &amp; -4 \\\\[1.1ex] 2 &amp; 1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b97f01989b5e9679f95d300cd64f3735_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} 3 &amp; 1 &amp; 2 \\\\[1.1ex] 2 &amp; 2 &amp; -4 \\\\[1.1ex] 2 &amp; 1 &amp; 0 \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"164\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77b38ebf03b8ed059edefd523c5ca1f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; 1  \\\\[1.1ex] 2 &amp; 2 \\end{vmatrix} = 6-2 = 4 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f174f72890bce94d148e1f6e88681ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = 2 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"310\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Abbiamo quindi trovato 2 casi in cui l&#8217;intervallo della matrice A varia con il valore che assume il parametro: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41e5cc7b6e9b3204f26e1c64e46f7057_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq 2 \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = 2\\ \\longrightarrow \\ Rg(A)=2}  \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"122\" width=\"304\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 2<\/h3>\n<p> Trovare l&#8217;intervallo della tabella seguente in base al valore del parametro <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5b28f21cc2e7211d9dae9b6685b541fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] a &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; a \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"150\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La matrice A avr\u00e0 al massimo rango 3, perch\u00e9 \u00e8 una matrice 3\u00d73. Pertanto, la prima cosa da fare \u00e8 risolvere il determinante dell&#8217;intera matrice (con la regola di Sarrus), per vedere se pu\u00f2 essere di rango 3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-30c8c16fea09001059a5d66727fc7be3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] a &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; a \\end{vmatrix} &amp; =2a-12-2a+2+12-2a^2 \\\\ &amp;=2-2a^2\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"335\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Impostiamo il risultato uguale a 0 per vedere quando l&#8217;array sar\u00e0 di rango 2 e quando di rango 3: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-28c4eeb004bd0bf3db692ee22c659a40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2-2a^2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"89\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e39820ff30d5df06ac09f254dcebeef0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2a^2=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3c5cad133a274f40a2151ad9e9310825_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a^2=\\cfrac{-2}{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"74\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8b5cd6314cc67aa83d49e16072e9314b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a^2=1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-63f049dc27947cfc24afdd331acefe23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a=\\pm 1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pertanto, quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e8 diverso da +1 e -1, il determinante 3\u00d73 sar\u00e0 diverso da 0 e, quindi, il rango della matrice sar\u00e0 3.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e7d26d825cd80ee861dd13168dafd408_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq +1, -1 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"354\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora vediamo cosa succede quando <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ee9005a2708f5bcb0f0fba0cefed3dfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=+1 :\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"65\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7b95d408f076c4978c8605380a277cdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = +1 \\longrightarrow A= \\begin{pmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] 1 &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fcd50b9549925b5011a6c20943c326ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] 1 &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; 1 \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"178\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03242296f208e07b9c4d634f0b7724cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}  2 &amp; 1 \\\\[1.1ex]  1 &amp; 3 \\end{vmatrix} = 6-1 = 5 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2550b439990981d1b74f72b1649a57e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = +1 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora vediamo cosa succede quando <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-53cc36b0e502c4e9a0aa575015035a8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=-1 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a2d16421400df26760d811229215ac83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = -1 \\longrightarrow A= \\begin{pmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] -1 &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"258\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ecd0b86cd6c59a0911f0c39ca7599806_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} 2 &amp; 2 &amp; 1 \\\\[1.1ex] -1 &amp; 1 &amp; 3 \\\\[1.1ex] -2 &amp; -2 &amp; -1  \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"191\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-378e43f0ef61ccabf82dacb5ac70466f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 2 &amp; 2  \\\\[1.1ex] -1 &amp; 1 \\end{vmatrix} =2-(-2) = 4 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d12346bae2f327e7e1ee6c5276a599cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = -1 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Abbiamo quindi trovato 3 casi in cui il range della matrice A varia a seconda del valore che assume il parametro: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6bf1904cee51914e041d94f588fed84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq +1,-1 \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = +1\\ \\longrightarrow \\ Rg(A)=2} \\\\[3ex] \\color{black} \\bm{a = -1\\ \\longrightarrow \\ Rg(A)=2}  \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"348\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 3<\/h3>\n<p> Calcola l&#8217;intervallo della tabella seguente in base al valore del parametro <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-090a99d3b4111785433e5c769589eb01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} a+1 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; a-3  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"184\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La matrice A avr\u00e0 al massimo rango 3, perch\u00e9 \u00e8 una matrice 3\u00d73. Pertanto, la prima cosa da fare \u00e8 risolvere il determinante dell&#8217;intera matrice (con la regola di Sarrus), per vedere se pu\u00f2 essere di rango 3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fec1cb52bb87fa2bccb40b70e1f21c7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} a+1 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; a-3 \\end{vmatrix} &amp; =(a+1)(a-3) +2+0-5+6(a+1)-0 \\\\ &amp; = a^2-3a+a-3 +2-5+6a+6 \\\\[1.5ex] &amp; =a^2+4a\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"468\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Impostiamo il risultato uguale a 0 per vedere quando l&#8217;array sar\u00e0 di rango 2 e quando di rango 3:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f8e26b9f10414656086a0c25d28ea04f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a^2+4a=0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"90\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Questa \u00e8 un&#8217;equazione quadratica incompleta, quindi estraiamo un fattore comune:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3f6035239798b59504a776dac1f0e21a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a(a+4)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E impostiamo ogni termine uguale a 0:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-43b38da320da538e46c6b4515de48568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a(a+4)=0 \\longrightarrow \\begin{cases} \\bm{a = 0} \\\\[2ex] a+4=0  \\ \\longrightarrow \\ \\bm{a=-4}\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"328\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Abbiamo ottenuto 0 e -4 come soluzioni. Pertanto, quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e8 diverso da 0 e -4, il determinante 3\u00d73 sar\u00e0 diverso da 0 e, quindi, il rango della matrice sar\u00e0 3.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-15908960ef2cfcd2105c4b901fb6cb49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq 0, -4 \\ \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"340\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora vediamo cosa succede quando <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d229e6228a70e103acbec8ca88c12d7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=0 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d97b25f01cb00d4677da0de5b4340ddb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = 0 \\longrightarrow A= \\begin{pmatrix} 1 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; -3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"230\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9e0f3c315588dff8274873001f727a69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} 1 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; -3 \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"178\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c132b22650c707d9f410c3d9c1e8da35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 1  \\\\[1.1ex] 0 &amp; 1 \\end{vmatrix} = 1-0 = 1 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d33aeec452b54112a958bfeadf014fe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = 0 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"310\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora vediamo cosa succede quando <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0287c5c8b769f316fb7d382ea3332fa7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=-4 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9ce7e40d9d78ecddc5ee81fc799c8767_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = -4 \\longrightarrow A= \\begin{pmatrix} -3 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; -7  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-52d15d0bceeb1dbbc415fb4825ce9a05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A \\end{vmatrix} = \\begin{vmatrix} -3 &amp; 1 &amp; -5 \\\\[1.1ex] 0 &amp; 1 &amp; -2 \\\\[1.1ex] -1 &amp; 3 &amp; -7 \\end{vmatrix}= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"178\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d45971f1dbda32405246de38bb68bd92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} -3 &amp; 1 \\\\[1.1ex] 0 &amp; 1\\end{vmatrix} =-3-0 = -3 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"213\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0551d4c8535193e378fc38c2e5580157_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = -4 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Abbiamo quindi trovato 3 casi in cui il range della matrice A varia a seconda del valore che assume il parametro: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-844f152985a2d84be1456501dfdc16e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq 0,-4 \\longrightarrow \\ Rg(A)=3} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = 0\\ \\longrightarrow \\ Rg(A)=2} \\\\[3ex] \\color{black} \\bm{a = -4\\ \\longrightarrow \\ Rg(A)=2}  \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"334\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 4<\/h3>\n<p> Trova l&#8217;estensione della seguente matrice di dimensione 3\u00d74 in base al valore del parametro <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a :\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4da7907bd0e8f80006ea47d2437b3f3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} -1&amp;-3&amp;-2&amp;1\\\\[1.1ex] 4&amp;12&amp;8&amp;-4\\\\[1.1ex] 2&amp;6&amp;4&amp;a \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"203\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>vedi soluzione<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> La matrice A avr\u00e0 al massimo rango 3, poich\u00e9 non possiamo calcolare alcun <a href=\"https:\/\/mathority.org\/it\/determinanti-4x4-con-esempi-complementari-ed-esercizi-risolti\/\">determinante 4\u00d74<\/a> . Pertanto, la prima cosa che dobbiamo fare \u00e8 risolvere tutti i possibili determinanti di ordine 3 (con la regola di Sarrus), per vedere se pu\u00f2 essere di ordine 3: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2db025b8ecf4323d4a912d84a215d8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} -1&amp;-3&amp;-2\\\\[1.1ex] 4&amp;12&amp;8\\\\[1.1ex] 2&amp;6&amp;4 \\end{vmatrix} &amp; =-48-48-48+48+48+48 =\\bm{0}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"395\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e40592bf6f8bfd13cb68a1fd0393cebb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} -1&amp;-3&amp;1\\\\[1.1ex] 4&amp;12&amp;-4\\\\[1.1ex] 2&amp;6&amp;a \\end{vmatrix} &amp; =-12a+24+24-24-24+12a=\\bm{0}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"414\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce1c28ae4120f0b37059b763e576d2eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} -1&amp;-2&amp;1\\\\[1.1ex] 4&amp;8&amp;-4\\\\[1.1ex] 2&amp;4&amp;a \\end{vmatrix} &amp; =-8a+16+16-16-16+8a=\\bm{0}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"396\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-668e9096b00b90ee4cc48d272b17e7bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} -3&amp;-2&amp;1\\\\[1.1ex] 12&amp;8&amp;-4\\\\[1.1ex] 6&amp;4&amp;a \\end{vmatrix} &amp; =-24a+48+48-48-48+24a=\\bm{0}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"414\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> I risultati di tutti i possibili determinanti di ordine 3 sono 0, qualunque sia il valore di<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Quindi la matrice non sar\u00e0 mai di rango 3, poich\u00e9 non importa quale valore assuma<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> che non ci sar\u00e0 mai un determinante 3\u00d73 diverso da 0.<\/p>\n<p class=\"has-text-align-left\"> Quindi ora proviamo i determinanti di dimensione 2 \u00d7 2. Tuttavia, anche tutti i determinanti di ordine 2 danno 0 tranne i seguenti:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c4408f1ccf562196943209356e50e892_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix} 8&amp;-4\\\\[1.1ex] 4&amp;a \\end{vmatrix} &amp; =8a+16 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"139\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora impostiamo il risultato uguale a 0 e risolviamo l&#8217;equazione: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5494bb524be48bc22a1cb054556c3a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 8a+16=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"90\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a6fc020bc84c4ba3f1989065a2207fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 8a=-16\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"74\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-02680bced4f2a76a7d23c5b9e6a2ecbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a=\\cfrac{-16}{8} =-2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"120\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pertanto, quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e8 diverso da -2, il determinante 2\u00d72 sar\u00e0 diverso da 0 e, quindi, il rango della matrice sar\u00e0 2.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-04b3447f6e823c3e11b66919654e7a5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black}\\phantom{33} \\bm{a \\neq -2 \\ \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ora vediamo cosa succede quando <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0e72cd3ad115f5d34fb5077b4d7d278a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a=-2 :\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"65\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ecbf63b188b46c05e67741cee83d7a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a = -2 \\longrightarrow A= \\begin{pmatrix} -1&amp;-3&amp;-2&amp;1\\\\[1.1ex] 4&amp;12&amp;8&amp;-4\\\\[1.1ex] 2&amp;6&amp;4&amp;-2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"297\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Come abbiamo visto in precedenza, quando<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1961b1513bd5718956433f1198aa5844_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e8 -2, tutti i determinanti di ordine 2 sono 0. Non pu\u00f2 quindi essere di rango 2. E poich\u00e9 esiste almeno un determinante 1\u00d71 diverso da 0, in questo caso il rango della matrice \u00e8 1:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eb1cee57ae9619b3e4fdbf2357893425_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\[-2ex] \\color{black} \\phantom{33} \\bm{a = -2 \\ \\longrightarrow \\ Rg(A)=1} \\phantom{33} \\\\[-2ex] &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Abbiamo quindi trovato 2 casi in cui l&#8217;intervallo della matrice A varia con il valore che assume il parametro: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2bdfb67894431a4a08a3e791dcda0313_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\color{blue} \\boxed{ \\begin{array}{c} \\\\ \\color{black} \\phantom{33} \\bm{a \\neq -2 \\longrightarrow \\ Rg(A)=2} \\phantom{33} \\\\[3ex] \\color{black} \\bm{a = -2\\ \\longrightarrow \\ Rg(A)=1}   \\\\ &amp; \\end{array} }\" title=\"Rendered by QuickLaTeX.com\" height=\"122\" width=\"317\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>In questa pagina vedrai come calcolare il rango di una tabella in base a un parametro. Troverai anche esempi passo passo ed esercizi risolti su come trovare l&#8217;intervallo di una matrice in base a un parametro. Per comprendere appieno la procedura per studiare il rango delle matrici con parametri, \u00e8 importante sapere gi\u00e0 come calcolare &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/estensione-di-una-matrice-in-funzione-di-un-parametro-esempi-ed-esercizi-risolti-di-matrici-2x2-3x3-3x4-4x4\/\"> <span class=\"screen-reader-text\">Intervallo di un array basato su un parametro<\/span> Leggi altro &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[4],"tags":[],"class_list":["post-299","post","type-post","status-publish","format-standard","hentry","category-calcolatrici"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Intervallo di un array in base a un parametro -<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/it\/estensione-di-una-matrice-in-funzione-di-un-parametro-esempi-ed-esercizi-risolti-di-matrici-2x2-3x3-3x4-4x4\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Intervallo di un array in base a un parametro -\" \/>\n<meta property=\"og:description\" content=\"In questa pagina vedrai come calcolare il rango di una tabella in base a un parametro. Troverai anche esempi passo passo ed esercizi risolti su come trovare l&#8217;intervallo di una matrice in base a un parametro. Per comprendere appieno la procedura per studiare il rango delle matrici con parametri, \u00e8 importante sapere gi\u00e0 come calcolare &hellip; Intervallo di un array basato su un parametro Leggi altro &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/it\/estensione-di-una-matrice-in-funzione-di-un-parametro-esempi-ed-esercizi-risolti-di-matrici-2x2-3x3-3x4-4x4\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T17:07:12+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png\" \/>\n<meta name=\"author\" content=\"Squadra di Mathority\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Scritto da\" \/>\n\t<meta name=\"twitter:data1\" content=\"Squadra di Mathority\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tempo di lettura stimato\" \/>\n\t<meta name=\"twitter:data2\" 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Per comprendere appieno la procedura per studiare il rango delle matrici con parametri, \u00e8 importante sapere gi\u00e0 come calcolare &hellip; Intervallo di un array basato su un parametro Leggi altro &raquo;","og_url":"https:\/\/mathority.org\/it\/estensione-di-una-matrice-in-funzione-di-un-parametro-esempi-ed-esercizi-risolti-di-matrici-2x2-3x3-3x4-4x4\/","article_published_time":"2023-07-06T17:07:12+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a48b1eabd1692d9c9da67cbdaef7db3c_l3.png"}],"author":"Squadra di Mathority","twitter_card":"summary_large_image","twitter_misc":{"Scritto da":"Squadra di Mathority","Tempo di lettura stimato":"5 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