{"id":340,"date":"2023-07-06T17:07:12","date_gmt":"2023-07-06T17:07:12","guid":{"rendered":"https:\/\/mathority.org\/nl\/omvang-van-een-matrix-als-functie-van-een-parameter-voorbeelden-en-opgeloste-oefeningen-van-matrices-2x2-3x3-3x4-4x4\/"},"modified":"2023-07-06T17:07:12","modified_gmt":"2023-07-06T17:07:12","slug":"omvang-van-een-matrix-als-functie-van-een-parameter-voorbeelden-en-opgeloste-oefeningen-van-matrices-2x2-3x3-3x4-4x4","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/omvang-van-een-matrix-als-functie-van-een-parameter-voorbeelden-en-opgeloste-oefeningen-van-matrices-2x2-3x3-3x4-4x4\/","title":{"rendered":"Bereik van een array op basis van een parameter"},"content":{"rendered":"<p>Op deze pagina ziet u hoe u de <strong>rangorde van een tabel kunt berekenen op basis van een parameter.<\/strong> Ook vindt u stapsgewijze voorbeelden en opgeloste oefeningen over hoe u het bereik van een matrix kunt vinden op basis van \u00e9\u00e9n parameter.<\/p>\n<p> Om de procedure voor het bestuderen van de rangorde van matrices met parameters volledig te begrijpen, is het belangrijk dat u al weet <a href=\"https:\/\/mathority.org\/nl\">hoe u de rangorde van een matrix op basis van determinanten kunt berekenen<\/a> . We raden u daarom aan eerst deze twee dingen te leren voordat u verder leest.<\/p>\n<h2 class=\"wp-block-heading\"> Hoe u het bereik van een array kunt berekenen op basis van een parameter. Voorbeeld:<\/h2>\n<ul>\n<li> Bepaalt het bereik van matrix A op basis van verschillende parameterwaarden\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> Matrix A heeft hoogstens rang 3, omdat het een matrix van orde 3 is. Daarom moeten we eerst <strong>de determinant van de gehele 3&#215;3-matrix oplossen<\/strong> met <a href=\"https:\/\/mathority.org\/nl\/determinanten-3x3-sarrusregelvoorbeelden-en-opgeloste-oefeningen\/\">de regel van Sarrus<\/a> , om te zien of deze rang 3 kan zijn:<\/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> Het resultaat van de determinant is een functie van de parameter<\/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>We stellen daarom het resultaat gelijk aan 0<\/strong> om te zien wanneer de tafel van rang 2 zal zijn en wanneer deze van rang 3 zal zijn:<\/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> En we lossen de resulterende vergelijking op: <\/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> Daarom, wanneer<\/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> Of het nu +1 of -1 is, de 3\u00d73 determinant zal 0 zijn en daarom zal de rangorde van de matrix niet 3 zijn. Aan de andere kant, wanneer<\/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> verschillend is van +1 en -1, zal de determinant verschillend zijn van 0 en daarom zal de matrix van rang 3 zijn.<\/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> Laten we nu eens kijken wat er wanneer gebeurt<\/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> Zoals we eerder zagen, wanneer<\/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> is 1, de determinant van de matrix is 0. Deze kan daarom niet van rang 3 zijn. We proberen nu een <a href=\"https:\/\/mathority.org\/nl\/determinanten-2x2-voorbeelden-en-opgeloste-oefeningen\/\">2\u00d72 determinant<\/a> te berekenen die verschilt van 0 binnen de matrix, bijvoorbeeld die van de linkerbovenhoek:<\/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> De determinant van orde 2 is anders dan 0. Dus als de parameter<\/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> of +1, de <strong>rangorde van de matrix is 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> Zodra we het bereik van de matrix zien wanneer<\/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> en wanneer<\/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> Laten we eens kijken wat er wanneer gebeurt<\/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> Zoals we in het begin zagen, wanneer<\/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 en de determinant van de matrix is 0. Daarom kan deze niet op rang 3 worden gezet. Daarom moeten we proberen een determinant van 2\u00d72 in de matrix tegen te komen die anders is dan 0, bijvoorbeeld de lagere onderdeel van de matrix. LINKS:<\/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> De determinant van dimensie 2 is anders dan 0. Dus als de parameter<\/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> of -1, de <strong>rangorde van de tafel is 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> We hebben daarom 3 verschillende gevallen gevonden waarin de rangorde van matrix A afhangt van de waarde die de parameter aanneemt<\/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> Hier is de <strong>samenvatting<\/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> Nu u weet hoe u de reeks parameterafhankelijke matrices moet bespreken, kunt u oefenen met de onderstaande stapsgewijze oefeningen. Om ze op te lossen, zullen de <a href=\"https:\/\/mathority.org\/nl\/eigenschappen-van-determinanten-voorbeelden-en-oefeningen-2x2-3x3\/\">eigenschappen van de determinatoren<\/a> je zeker helpen, dus als je er niet erg duidelijk over bent, raad ik je aan eerst een kijkje te nemen op de gelinkte pagina, waar ze allemaal met voorbeelden worden uitgelegd.<\/p>\n<h2 class=\"wp-block-heading\"> Problemen met op parameters gebaseerde matrixbereiken opgelost<\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Bestudeer het bereik van de volgende tabel op basis van de parameterwaarde <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Matrix A heeft maximaal rang 3, omdat het een 3\u00d73-matrix is. Daarom moeten we eerst de determinant van de hele matrix oplossen (met de regel van Sarrus), om te zien of deze van rang 3 kan zijn:<\/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\"> We stellen het resultaat gelijk aan 0 om te zien wanneer de array rang 2 zal zijn en wanneer rang 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\"> Daarom, wanneer<\/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> verschillend is van 2, zal de determinant 3\u00d73 anders zijn dan 0 en daarom zal de rangorde van de matrix 3 zijn.<\/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\"> Laten we nu eens kijken wat er wanneer gebeurt <\/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\"> We hebben daarom 2 gevallen gevonden waarin het bereik van matrix A varieert met de waarde die de parameter aanneemt: <\/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\">Oefening 2<\/h3>\n<p> Zoek het bereik van de volgende tabel op basis van de parameterwaarde <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Matrix A heeft maximaal rang 3, omdat het een 3\u00d73-matrix is. Daarom moeten we eerst de determinant van de hele matrix oplossen (met de regel van Sarrus), om te zien of deze van rang 3 kan zijn:<\/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\"> We stellen het resultaat gelijk aan 0 om te zien wanneer de array rang 2 zal zijn en wanneer rang 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\"> Daarom, wanneer<\/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> verschillend is van +1 en -1, zal de 3\u00d73-determinant anders zijn dan 0 en daarom zal de rangorde van de matrix 3 zijn.<\/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\"> Laten we nu eens kijken wat er wanneer gebeurt <\/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\"> Laten we nu eens kijken wat er wanneer gebeurt <\/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\"> We hebben daarom 3 gevallen gevonden waarin het bereik van matrix A varieert afhankelijk van de waarde die de parameter aanneemt: <\/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\"> Oefening 3<\/h3>\n<p> Berekent het bereik van de volgende tabel op basis van de parameterwaarde <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Matrix A heeft maximaal rang 3, omdat het een 3\u00d73-matrix is. Daarom moeten we eerst de determinant van de hele matrix oplossen (met de regel van Sarrus), om te zien of deze van rang 3 kan zijn:<\/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\"> We stellen het resultaat gelijk aan 0 om te zien wanneer de array rang 2 zal zijn en wanneer rang 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\"> Dit is een onvolledige kwadratische vergelijking, dus we halen er een gemeenschappelijke factor uit:<\/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\"> En we stellen elke term gelijk aan 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\"> We verkregen 0 en -4 als oplossingen. Daarom, wanneer<\/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> verschillend is van 0 en -4, zal de 3\u00d73-determinant anders zijn dan 0 en daarom zal de rangorde van de matrix 3 zijn.<\/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\"> Laten we nu eens kijken wat er wanneer gebeurt <\/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\"> Laten we nu eens kijken wat er wanneer gebeurt <\/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\"> We hebben daarom 3 gevallen gevonden waarin het bereik van matrix A varieert afhankelijk van de waarde die de parameter aanneemt: <\/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\">Oefening 4<\/h3>\n<p> Zoek de omvang van de volgende matrix met afmeting 3\u00d74 volgens de waarde van de parameter <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De matrix A zal hoogstens rang 3 zijn, aangezien we geen <a href=\"https:\/\/mathority.org\/nl\/4x4-determinanten-door-complementaire-voorbeelden-en-opgeloste-oefeningen\/\">4\u00d74 determinant<\/a> kunnen berekenen. Daarom moeten we eerst alle mogelijke determinanten van orde 3 oplossen (met de regel van Sarrus), om te zien of deze van orde 3 kunnen zijn: <\/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\"> De resultaten van alle mogelijke determinanten van orde 3 zijn 0, ongeacht de waarde ervan<\/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> . De matrix zal dus nooit rang 3 hebben, omdat het niet uitmaakt welke waarde deze heeft<\/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> dat er nooit een andere 3\u00d73 determinant dan 0 zal zijn.<\/p>\n<p class=\"has-text-align-left\"> Dus nu proberen we determinanten van dimensie 2 \u00d7 2. Alle determinanten van orde 2 geven echter ook 0, behalve het volgende:<\/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\"> Nu stellen we het resultaat gelijk aan 0 en lossen we de vergelijking op: <\/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\"> Daarom, wanneer<\/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> verschillend is van -2, zal de determinant 2\u00d72 verschillend zijn van 0 en daarom zal de rangorde van de matrix 2 zijn.<\/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\"> Laten we nu eens kijken wat er wanneer gebeurt <\/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\"> Zoals we eerder zagen, wanneer<\/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> is -2, alle determinanten van orde 2 zijn 0. Deze kan daarom niet van rang 2 zijn. En aangezien er minstens \u00e9\u00e9n 1\u00d71 determinant bestaat die verschilt van 0, is in dit geval de rang van de matrix 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\"> We hebben daarom 2 gevallen gevonden waarin het bereik van matrix A varieert met de waarde die de parameter aanneemt: <\/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>Op deze pagina ziet u hoe u de rangorde van een tabel kunt berekenen op basis van een parameter. Ook vindt u stapsgewijze voorbeelden en opgeloste oefeningen over hoe u het bereik van een matrix kunt vinden op basis van \u00e9\u00e9n parameter. Om de procedure voor het bestuderen van de rangorde van matrices met parameters &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/omvang-van-een-matrix-als-functie-van-een-parameter-voorbeelden-en-opgeloste-oefeningen-van-matrices-2x2-3x3-3x4-4x4\/\"> <span class=\"screen-reader-text\">Bereik van een array op basis van een parameter<\/span> Lees meer &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":[36],"tags":[],"class_list":["post-340","post","type-post","status-publish","format-standard","hentry","category-rekenmachines"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Bereik van een array op basis van een parameter -<\/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\/nl\/omvang-van-een-matrix-als-functie-van-een-parameter-voorbeelden-en-opgeloste-oefeningen-van-matrices-2x2-3x3-3x4-4x4\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bereik van een array op basis van een parameter -\" \/>\n<meta property=\"og:description\" content=\"Op deze pagina ziet u hoe u de rangorde van een tabel kunt berekenen op basis van een parameter. 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Ook vindt u stapsgewijze voorbeelden en opgeloste oefeningen over hoe u het bereik van een matrix kunt vinden op basis van \u00e9\u00e9n parameter. 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