{"id":297,"date":"2023-07-06T17:31:42","date_gmt":"2023-07-06T17:31:42","guid":{"rendered":"https:\/\/mathority.org\/it\/rango-di-una-matrice\/"},"modified":"2023-07-06T17:31:42","modified_gmt":"2023-07-06T17:31:42","slug":"rango-di-una-matrice","status":"publish","type":"post","link":"https:\/\/mathority.org\/it\/rango-di-una-matrice\/","title":{"rendered":"Calcolare il rango di una matrice in base ai determinanti"},"content":{"rendered":"<p>In questa pagina vedrai cos&#8217;\u00e8 e come calcolare l&#8217; <strong>intervallo di una matrice<\/strong> in base ai determinanti. Inoltre troverai esempi ed esercizi risolti per imparare a trovare facilmente l&#8217;estensione di una matrice. Inoltre, vedrai anche le propriet\u00e0 dell&#8217;intervallo di una matrice.<\/p>\n<h2 class=\"wp-block-heading\"> Qual \u00e8 il rango di una matrice?<\/h2>\n<p> La definizione di intervallo di una matrice \u00e8:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Il <strong>rango di una matrice<\/strong> \u00e8 l&#8217;ordine della sottomatrice quadrata pi\u00f9 grande il cui determinante \u00e8 diverso da 0.<\/p>\n<p> In questa pagina impareremo l&#8217;intervallo di una matrice con il metodo dei determinanti, ma l&#8217;intervallo di una matrice pu\u00f2 essere determinato anche con il metodo gaussiano, sebbene sia pi\u00f9 lento e complicato.<\/p>\n<p> Una volta che sappiamo qual \u00e8 l&#8217;intervallo di una matrice, vedremo come trovarlo in base ai determinanti. Ma tieni presente che per risolvere l&#8217;estensione di una matrice, devi prima sapere come calcolare <a href=\"https:\/\/mathority.org\/it\/determinanti-3x3-esempi-di-regole-sarrus-ed-esercizi-risolti\/\">3&#215;3 determinanti<\/a> .<\/p>\n<h2 class=\"wp-block-heading\"> Come conoscere l&#8217;estensione di una matrice? Esempio:<\/h2>\n<ul>\n<li> Calcola l\u2019estensione della seguente matrice di dimensione 3\u00d74: <\/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-79e80ea42079a394262a4fcce5a863f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{cccc} 1 &amp; 3 &amp; 4 &amp; -1 \\\\[1.1ex] 0 &amp; 2 &amp; 1 &amp; -1  \\\\[1.1ex] 3 &amp; -1 &amp; 7 &amp; 2 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"191\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Inizieremo sempre cercando di vedere se la matrice ha il rango massimo risolvendo il determinante pi\u00f9 grande dell&#8217;ordine. E, se il determinante di questo ordine \u00e8 uguale a 0, continueremo a testare i determinanti di ordine inferiore finch\u00e9 non ne troveremo uno diverso da 0.<\/p>\n<p> In questo caso si tratta di una matrice di dimensione 3\u00d74. <strong>Sar\u00e0 quindi al massimo di rango 3<\/strong> , poich\u00e9 non possiamo creare alcun determinante di ordine 4. Quindi prendiamo una sottomatrice qualsiasi 3\u00d73 e vediamo se il suo determinante \u00e8 0. Ad esempio, risolviamo il determinante delle prime 3 colonne con la regola di Sarrus:<\/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-819aaaa272025ce70b7852d00680483d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; \\cellcolor[HTML]{ABEBC6}3 &amp; \\cellcolor[HTML]{ABEBC6}4  &amp; -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp; \\cellcolor[HTML]{ABEBC6}2 &amp; \\cellcolor[HTML]{ABEBC6} 1 &amp; -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\\\[-2ex]\\cellcolor[HTML]{ABEBC6} 3 &amp;\\cellcolor[HTML]{ABEBC6} -1 &amp; \\cellcolor[HTML]{ABEBC6} 7 &amp; 2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"570\" 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-aedcd597b0cd9cd0ad11ab1d99bd0e5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 3 &amp; 4 \\\\[1.1ex] 0 &amp; 2 &amp; 1   \\\\[1.1ex] 3 &amp; -1 &amp; 7  \\end{vmatrix} = 14 + 9 + 0 - 24 + 1 - 0 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"318\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Il determinante delle colonne 1, 2 e 3 \u00e8 0. Dobbiamo ora provare un altro determinante, ad esempio quello delle colonne 1, 2 e 4:<\/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-ddfbcde7994d5665983fda2423c82de3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; \\cellcolor[HTML]{ABEBC6} 3 &amp; 4  &amp; \\cellcolor[HTML]{ABEBC6}-1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp; &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp;\\cellcolor[HTML]{ABEBC6}2 &amp; 1 &amp; \\cellcolor[HTML]{ABEBC6} -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6}&amp; &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex]\\cellcolor[HTML]{ABEBC6} 3 &amp; \\cellcolor[HTML]{ABEBC6}-1 &amp; 7 &amp; \\cellcolor[HTML]{ABEBC6}2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"565\" 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-6f13263d4697369ed7d98bf7f972d15f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 3 &amp; -1 \\\\[1.1ex] 0 &amp; 2 &amp; -1   \\\\[1.1ex] 3 &amp; -1 &amp; 2  \\end{vmatrix} = 4 -9 + 0 + 6-1 - 0 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"314\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ci ha dato anche 0. Continuiamo quindi a testare i determinanti di ordine 3 per vedere se ce ne sono altri oltre a 0. Testiamo ora il determinante formato dalle colonne 1, 3 e 4:<\/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-2682212fc905820bb8c2c2b73eeb49e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; 3 &amp; \\cellcolor[HTML]{ABEBC6}4  &amp; \\cellcolor[HTML]{ABEBC6}-1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp;2 &amp; \\cellcolor[HTML]{ABEBC6} 1 &amp; \\cellcolor[HTML]{ABEBC6} -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp;  &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex]\\cellcolor[HTML]{ABEBC6} 3 &amp;  -1 &amp; \\cellcolor[HTML]{ABEBC6} 7 &amp; \\cellcolor[HTML]{ABEBC6}2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"570\" 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-c84fbf30f1005e0bdd6496369c68efb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; 4 &amp; -1 \\\\[1.1ex] 0 &amp; 1 &amp; -1   \\\\[1.1ex] 3 &amp; 7 &amp; 2  \\end{vmatrix} = 2 -12+0 +3 +7- 0 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"309\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Tra i determinanti di ordine 3 provare semplicemente il determinante composto dalle colonne 2, 3 e 4:<\/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-610e7befed3409c44ad1b84a6c84605d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}1 &amp; \\cellcolor[HTML]{ABEBC6}3 &amp; \\cellcolor[HTML]{ABEBC6}4  &amp; \\cellcolor[HTML]{ABEBC6}-1 \\\\  &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex] 0 &amp; \\cellcolor[HTML]{ABEBC6}2 &amp; \\cellcolor[HTML]{ABEBC6} 1 &amp; \\cellcolor[HTML]{ABEBC6} -1 \\\\  &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} \\\\[-2ex] 3 &amp; \\cellcolor[HTML]{ABEBC6} -1 &amp; \\cellcolor[HTML]{ABEBC6} 7 &amp; \\cellcolor[HTML]{ABEBC6}2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"570\" 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-6377c641072d9eba07fd2b9670ffbf50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle   \\begin{vmatrix} 3 &amp; 4 &amp; -1 \\\\[1.1ex]  2 &amp; 1 &amp; -1  \\\\[1.1ex] -1 &amp; 7 &amp; 2 \\end{vmatrix} = 6+4-14-1+21-16 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"341\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Abbiamo gi\u00e0 provato tutti i possibili determinanti 3&#215;3 della matrice A, e poich\u00e9 nessuno di questi \u00e8 diverso da 0, <strong>la matrice non \u00e8 di rango 3<\/strong> . Pertanto, al massimo sar\u00e0 il rango 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-157fd11377c30ccf66e64960e295866b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A) < 3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Vedremo ora se la matrice \u00e8 di rango 2. Per fare ci\u00f2, dobbiamo trovare una sottomatrice quadrata di ordine 2 il cui determinante sia diverso da 0. Proveremo la sottomatrice 2\u00d72 nell&#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-b0ae4ab76e4e45bbb1aecd49af2523a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; \\cellcolor[HTML]{ABEBC6}3 &amp; 4  &amp; -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp; &amp; \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp; \\cellcolor[HTML]{ABEBC6}2 &amp;  1 &amp; -1 &amp;  &amp; &amp; \\\\[-2ex] 3 &amp; -1 &amp;  7 &amp; 2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"75\" width=\"411\" 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-3320ea7301733c03681caf31e7539b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 3 \\\\[1.1ex] 0 &amp; 2  \\end{vmatrix} = 2-0 = 2 \\bm{ \\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"167\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Abbiamo trovato un determinante di ordine 2 diverso da 0 all&#8217;interno della matrice. Di conseguenza <strong>la matrice \u00e8 di rango 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Risolti i problemi relativi all&#8217;ambito della matrice<\/h2>\n<h3 class=\"wp-block-heading\"> Esercizio 1<\/h3>\n<p> Determinare il rango della seguente matrice 2\u00d72: <\/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-ca5f88e86382a14720247e910084095c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; 1 \\\\[1.1ex] 5 &amp; 6  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" 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\"> Per prima cosa calcoliamo il determinante dell&#8217;intera matrice:<\/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-88be02e3f0e84b30178b811354994424_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}=\\begin{vmatrix} 3 &amp; 1 \\\\[1.1ex] 5 &amp; 6 \\end{vmatrix} = 18-5 = 13 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"233\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Abbiamo trovato un determinante di ordine 2 diverso da 0. Pertanto <strong>la matrice \u00e8 di rango 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/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> Trova l\u2019estensione della seguente matrice di dimensione 2\u00d72: <\/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-19dde855da87ad73bdec3135fca04e78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 3 \\\\[1.1ex] 4 &amp; 6  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" 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\"> Innanzitutto, risolviamo il determinante dell&#8217;intera matrice:<\/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-eb383f77013e752e0f22ad582dbd3c80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}=\\begin{vmatrix} 2 &amp; 3 \\\\[1.1ex] 4 &amp; 6 \\end{vmatrix} = 12-12 \\bm{=0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"201\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> L\u2019unico determinante 2\u00d72 possibile d\u00e0 0, quindi la matrice non \u00e8 di rango 2.<\/p>\n<p class=\"has-text-align-left\"> Ma all&#8217;interno della matrice ci sono determinanti 1&#215;1 diversi da 0, ad esempio:<\/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-9bfea6551282e7213ca85662eb657b6d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2  \\end{vmatrix} = 2 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"79\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>La matrice \u00e8 quindi di rango 1.<\/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-e9b93965a2d6e8834b62367fbe854e02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=1}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/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> Qual \u00e8 l&#8217;estensione della seguente matrice quadrata 3&#215;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-fbe69cc53a58fd72117fa4aaa7a0ec38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; -3 &amp; 2 \\\\[1.1ex] 2 &amp; 1 &amp; 4 \\\\[1.1ex] 1 &amp; 4 &amp; 2 \\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\"> Innanzitutto si calcola il determinante dell\u2019intera matrice con la regola di Sarrus:<\/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-a1bda19a46e006dfc43ade0e92f189e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix} 1 &amp; -3 &amp; 2 \\\\[1.1ex] 2 &amp; 1 &amp; 4 \\\\[1.1ex] 1 &amp; 4 &amp; 2 \\end{vmatrix} = 2-12+16-2-16+12 \\bm{=0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"380\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> L\u2019unico determinante 3\u00d73 possibile d\u00e0 0, quindi la matrice non \u00e8 di rango 3.<\/p>\n<p class=\"has-text-align-left\"> Ma all&#8217;interno della matrice ci sono determinanti di ordine 2 diversi da 0, ad esempio:<\/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-9a1e82c35249f351ba9513437da95c65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 \\\\[1.1ex] 2 &amp; 1  \\end{vmatrix} = 1 +6 = 7 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"181\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pertanto <strong>la matrice \u00e8 di rango 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/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> Calcolare il rango della seguente matrice 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-7d952325e084adb3fa3b97c7fc10c1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; -1 &amp; 1 \\\\[1.1ex] 4 &amp; -2 &amp; 3 \\\\[1.1ex] 2 &amp; 5 &amp; 2 \\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\"> Innanzitutto il determinante dell\u2019intera matrice si risolve con la regola di Sarrus:<\/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-819e9bea5c6d6d536a4dafba325ae45e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix} 3 &amp; -1 &amp; 1 \\\\[1.1ex] 4 &amp; -2 &amp; 3 \\\\[1.1ex] 2 &amp; 5 &amp; 2 \\end{vmatrix} = -12-6+20+4-45+8 =  -31\\bm{ \\neq0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"440\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il determinante dell&#8217;intera matrice vale qualcosa di diverso da 0. Pertanto, la matrice ha rango massimo, cio\u00e8 <strong>rango 3.<\/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-8548c1b53b3e4fbb5509589cb60f87b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=3}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 5<\/h3>\n<p> Qual \u00e8 il rango della seguente matrice 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-d90091dd51727e806e6788a9594735ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 5 &amp; -1 \\\\[1.1ex] 3 &amp; -2 &amp; -4 \\\\[1.1ex] 5 &amp; 3 &amp; -5 \\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\"> Innanzitutto si calcola il determinante dell\u2019intera matrice con la regola di Sarrus:<\/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-e4eee911dcf234c3fa63177e533901af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix}2 &amp; 5 &amp; -1 \\\\[1.1ex] 3 &amp; -2 &amp; -4 \\\\[1.1ex] 5 &amp; 3 &amp; -5 \\end{vmatrix} =20-100-9-10+24+75 \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"411\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> L\u2019unico determinante 3\u00d73 possibile d\u00e0 0, quindi la matrice non \u00e8 di rango 3.<\/p>\n<p class=\"has-text-align-left\"> Ma all\u2019interno della matrice ci sono determinanti 2\u00d72 diversi da 0, come ad esempio:<\/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-1397b7f935df1c8cce082c3f2f1418d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; 5 \\\\[1.1ex] 3 &amp; -2  \\end{vmatrix} = -4-15 = -19\\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"226\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>La matrice \u00e8 quindi di rango 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Esercizio 6<\/h3>\n<p> Trova l&#8217;estensione della seguente matrice 3&#215;4: <\/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-46ff20ee9ee9e4fac3e8858c55961f8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 3 &amp; 2 &amp; -4 &amp; 1 \\\\[1.1ex] 2 &amp; -2 &amp; -3 &amp; 5 \\\\[1.1ex] 5 &amp; 0 &amp; -7 &amp; 3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"175\" 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 non pu\u00f2 essere di rango 4, perch\u00e9 non possiamo creare determinanti 4\u00d74. Vediamo quindi se \u00e8 di rango 3 calcolando i determinanti 3\u00d73:<\/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-2897851f49a9556fc03aded5f1495297_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix}3 &amp; 2 &amp; -4  \\\\[1.1ex] 2 &amp; -2 &amp; -3  \\\\[1.1ex] 5 &amp; 0 &amp; -7 \\end{vmatrix} =42-30+0-40-0+28 \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"393\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il determinante delle prime 3 colonne d\u00e0 0. Tuttavia, il determinante delle ultime 3 colonne d\u00e0 qualcosa di diverso da 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-c0821770a710807269d81fb1f8dd21a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; -4 &amp; 1 \\\\[1.1ex] -2 &amp; -3 &amp; 5 \\\\[1.1ex] 0 &amp; -7 &amp; 3  \\end{vmatrix} = -18+0+14-0+70-24 = 42 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"400\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Quindi, poich\u00e9 al suo interno \u00e8 presente una sottomatrice di ordine 3 il cui determinante \u00e8 diverso da 0, <strong>la matrice \u00e8 di rango 3<\/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-8548c1b53b3e4fbb5509589cb60f87b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=3}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 7<\/h3>\n<p> Calcola l&#8217;intervallo della seguente matrice 4&#215;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-83e7cebc0d95d73f653cf54bd316c4f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 3 &amp; 4 &amp; -5  \\\\[1.1ex] 5 &amp; -2 &amp; -9  \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"164\" 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 non pu\u00f2 essere di rango 4, poich\u00e9 non \u00e8 possibile risolvere alcun determinante 4\u00d74. Vediamo quindi se \u00e8 di rango 3 eseguendo tutte le possibili determinanti 3&#215;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-1eec1befc1515b4405529ede01c55618_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 3 &amp; 4 &amp; -5 \\\\[1.1ex] 5 &amp; -2 &amp; -9\\end{vmatrix} \\bm{= 0} \\qquad \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 5 &amp; -2 &amp; -9 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"308\" 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-fa1de344c0bb747c9861afd4de5fa7c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 3 &amp; 4 &amp; -5 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0} \\qquad \\begin{vmatrix} 3 &amp; 4 &amp; -5 \\\\[1.1ex] 5 &amp; -2 &amp; -9 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"322\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Poich\u00e9 tutti i possibili determinanti 3\u00d73 danno 0, anche la matrice non \u00e8 di rango 3. Proviamo ora i determinanti 2\u00d72:<\/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-981d085760dd1b1dd46aab17f1d7ba78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 \\\\[1.1ex] 3 &amp; 4  \\end{vmatrix} =13 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"127\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Poich\u00e9 all&#8217;interno della matrice A esiste una sottomatrice di ordine 2 il cui determinante \u00e8 diverso da 0, <strong>la matrice \u00e8 di rango 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-c40408b072a81f61800b6521c3ede2cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Esercizio 8<\/h3>\n<p> Trova l&#8217;intervallo della seguente matrice 4 \u00d7 4: <\/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-bd5abef80b8d6ae74d4d60a0cf11e3ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} 2 &amp; 0 &amp; 1 &amp; -1  \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; -1  \\\\[1.1ex] 4 &amp; -2 &amp; -1 &amp; 3  \\\\[1.1ex] -1 &amp; 3 &amp; 2 &amp;  -4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" 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\"> Dobbiamo risolvere il determinante dell&#8217;intera matrice per vedere se \u00e8 di rango 4.<\/p>\n<p class=\"has-text-align-left\"> E per risolvere il determinante 4&#215;4, devi prima eseguire operazioni con le righe per trasformare tutti gli elementi di una colonna tranne uno in zero:<\/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-27642038b0dc0358b382aaeab5c55263_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 2 &amp; 0 &amp; 1 &amp; -1 \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; -1 \\\\[1.1ex] 4 &amp; -2 &amp; -1 &amp; 3 \\\\[1.1ex] -1 &amp; 3 &amp; 2 &amp; -4 \\end{vmatrix} \\begin{matrix} \\\\[1.1ex]  \\\\[1.1ex]\\xrightarrow{f_3 + 2f_2} \\\\[1.1ex] \\xrightarrow{f_4 - 3f_2} \\end{matrix} \\begin{vmatrix} 2 &amp; 0 &amp; 1 &amp; -1 \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; -1 \\\\[1.1ex] 10 &amp; 0 &amp; 1 &amp; 1 \\\\[1.1ex] -10 &amp; 0 &amp; -1 &amp; -1 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"111\" width=\"360\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo ora il determinante per deputati:<\/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-239ee8aebdd8161e1e86d3d093ade490_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 2 &amp; 0 &amp; 1 &amp; -1 \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; -1 \\\\[1.1ex] 10 &amp; 0 &amp; 1 &amp; 1 \\\\[1.1ex] -10 &amp; 0 &amp; -1 &amp; -1 \\end{vmatrix} \\displaystyle = 0\\bm{\\cdot} \\text{Adj(0)} +1\\bm{\\cdot} \\text{Adj(1)} +0\\bm{\\cdot} \\text{Adj(0)} + 0\\bm{\\cdot} \\text{Adj(0)}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"492\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Semplifichiamo i termini: <\/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-cd7140ff98995310b9c70e27c89dba05_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"=\\cancel{0\\bm{\\cdot} \\text{Adj(0)}}+1\\bm{\\cdot} \\text{Adj(1)} +\\cancel{0\\bm{\\cdot} \\text{Adj(0)}} + \\cancel{0\\bm{\\cdot} \\text{Adj(0)}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"343\" style=\"vertical-align: -5px;\"><\/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-029594698d2ffb9e165ed06c51bd495e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"= \\text{Adj(1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Calcoliamo l&#8217;aggiunto di 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-b30414c1569334502b1f17ee5380bd4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = (-1)^{2+2} \\begin{vmatrix}2 &amp;  1 &amp; -1 \\\\[1.1ex] 10 &amp; 1 &amp; 1 \\\\[1.1ex] -10 &amp; -1 &amp; -1\\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"202\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, infine, calcoliamo il determinante 3\u00d73 con la regola di Sarrus e la calcolatrice: <\/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-15b1857a0769672f75e0ba922e34413a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = (-1)^{4} \\cdot \\bigl[-2-10+10-10+2+10 \\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"298\" style=\"vertical-align: -7px;\"><\/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-f1b44afd86030388f2b3eb74f2117708_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = 1 \\cdot \\bigl[0 \\bigr]\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"61\" style=\"vertical-align: -7px;\"><\/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-e0f707524d15b7f3351b2e331ca447cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Il determinante 4&#215;4 dell&#8217;intera matrice d\u00e0 0, quindi la matrice A non sar\u00e0 di rango 4. Quindi ora vediamo se ha un determinante 3&#215;3 diverso da 0 al suo interno:<\/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-8a8dabdc8197de8102d9e0c50db837a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; 0 &amp; 1  \\\\[1.1ex] 3 &amp; 1 &amp; 1  \\\\[1.1ex] 4 &amp; -2 &amp; -1  \\end{vmatrix} = -2+0-6-4+4-0=8 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"355\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>La matrice A \u00e8 quindi di rango 3:<\/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-8548c1b53b3e4fbb5509589cb60f87b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{rg(A)=3}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"estil_titol_H2 wp-block-heading\"> Propriet\u00e0 dell&#8217;intervallo della matrice<\/h2>\n<ul>\n<li> L&#8217;intervallo non viene modificato se eliminiamo una riga riempita con zeri, una colonna o una riga riempita con 0.<\/li>\n<\/ul>\n<ul>\n<li> L&#8217;intervallo di una matrice non cambia se cambiamo l&#8217;ordine di due righe parallele, siano esse righe o colonne.<\/li>\n<\/ul>\n<ul>\n<li> Il rango di una matrice \u00e8 uguale a quello della sua trasposta.<\/li>\n<\/ul>\n<ul>\n<li> Se moltiplichi una riga o una colonna per un numero diverso da 0, il rango della matrice non cambia.<\/li>\n<\/ul>\n<ul>\n<li> La gamma di una tonalit\u00e0 non cambia quando eliminiamo una linea (riga o colonna) che \u00e8 una combinazione lineare di altre linee ad essa parallele.<\/li>\n<\/ul>\n<ul>\n<li> L&#8217;intervallo di una matrice non cambia se aggiungiamo altre righe parallele a una qualsiasi delle righe (righe o colonne) moltiplicate per un numero qualsiasi. Questo \u00e8 il motivo per cui il rango di una matrice pu\u00f2 essere calcolato anche con il metodo gaussiano. <\/li>\n<\/ul>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>In questa pagina vedrai cos&#8217;\u00e8 e come calcolare l&#8217; intervallo di una matrice in base ai determinanti. Inoltre troverai esempi ed esercizi risolti per imparare a trovare facilmente l&#8217;estensione di una matrice. Inoltre, vedrai anche le propriet\u00e0 dell&#8217;intervallo di una matrice. Qual \u00e8 il rango di una matrice? La definizione di intervallo di una matrice &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/it\/rango-di-una-matrice\/\"> <span class=\"screen-reader-text\">Calcolare il rango di una matrice in base ai determinanti<\/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":[7],"tags":[],"class_list":["post-297","post","type-post","status-publish","format-standard","hentry","category-determinante-di-una-matrice"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Calcolare il rango di una matrice in base ai determinanti - Mathority<\/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\/rango-di-una-matrice\/\" \/>\n<meta property=\"og:locale\" content=\"it_IT\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Calcolare il rango di una matrice in base ai determinanti - Mathority\" \/>\n<meta property=\"og:description\" content=\"In questa pagina vedrai cos&#8217;\u00e8 e come calcolare l&#8217; intervallo di una matrice in base ai determinanti. Inoltre troverai esempi ed esercizi risolti per imparare a trovare facilmente l&#8217;estensione di una matrice. Inoltre, vedrai anche le propriet\u00e0 dell&#8217;intervallo di una matrice. Qual \u00e8 il rango di una matrice? La definizione di intervallo di una matrice &hellip; Calcolare il rango di una matrice in base ai determinanti Leggi altro &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/it\/rango-di-una-matrice\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T17:31:42+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-79e80ea42079a394262a4fcce5a863f7_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\" content=\"5 minuti\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/it\/rango-di-una-matrice\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/it\/rango-di-una-matrice\/\"},\"author\":{\"name\":\"Squadra di Mathority\",\"@id\":\"https:\/\/mathority.org\/it\/#\/schema\/person\/8d6f69ffbe48aea8b43675a9a3ddb9c8\"},\"headline\":\"Calcolare il rango di una matrice in base ai determinanti\",\"datePublished\":\"2023-07-06T17:31:42+00:00\",\"dateModified\":\"2023-07-06T17:31:42+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/it\/rango-di-una-matrice\/\"},\"wordCount\":1014,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/it\/#organization\"},\"articleSection\":[\"Determinante di una matrice\"],\"inLanguage\":\"it-IT\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/it\/rango-di-una-matrice\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/it\/rango-di-una-matrice\/\",\"url\":\"https:\/\/mathority.org\/it\/rango-di-una-matrice\/\",\"name\":\"Calcolare il rango di una matrice in base ai determinanti - 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Mathority","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/it\/rango-di-una-matrice\/","og_locale":"it_IT","og_type":"article","og_title":"Calcolare il rango di una matrice in base ai determinanti - Mathority","og_description":"In questa pagina vedrai cos&#8217;\u00e8 e come calcolare l&#8217; intervallo di una matrice in base ai determinanti. Inoltre troverai esempi ed esercizi risolti per imparare a trovare facilmente l&#8217;estensione di una matrice. Inoltre, vedrai anche le propriet\u00e0 dell&#8217;intervallo di una matrice. Qual \u00e8 il rango di una matrice? 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