{"id":338,"date":"2023-07-06T17:31:42","date_gmt":"2023-07-06T17:31:42","guid":{"rendered":"https:\/\/mathority.org\/nl\/rang-van-een-matrix\/"},"modified":"2023-07-06T17:31:42","modified_gmt":"2023-07-06T17:31:42","slug":"rang-van-een-matrix","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/rang-van-een-matrix\/","title":{"rendered":"Bereken de rangorde van een matrix op basis van determinanten"},"content":{"rendered":"<p>Op deze pagina ziet u wat het is en hoe u het <strong>bereik van een matrix<\/strong> aan de hand van determinanten kunt berekenen. Daarnaast vind je voorbeelden en opgeloste oefeningen om te leren hoe je eenvoudig de omvang van een matrix kunt vinden. Bovendien ziet u ook de bereikeigenschappen van een matrix.<\/p>\n<h2 class=\"wp-block-heading\"> Wat is de rangorde van een matrix?<\/h2>\n<p> De bereikdefinitie van een matrix is:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> De <strong>rangorde van een matrix<\/strong> is de volgorde van de grootste vierkante submatrix waarvan de determinant verschillend is van 0.<\/p>\n<p> Op deze pagina zullen we leren over het bereik van een matrix met behulp van de methode van determinanten, maar het bereik van een matrix kan ook worden bepaald met de Gaussische methode, hoewel deze langzamer en ingewikkelder is.<\/p>\n<p> Zodra we weten wat het bereik van een matrix is, zullen we zien hoe we het bereik van een matrix kunnen vinden aan de hand van determinanten. Maar houd er rekening mee dat u, om de omvang van een matrix op te lossen, eerst moet weten hoe u <a href=\"https:\/\/mathority.org\/nl\/determinanten-3x3-sarrusregelvoorbeelden-en-opgeloste-oefeningen\/\">3&#215;3 determinanten<\/a> berekent.<\/p>\n<h2 class=\"wp-block-heading\"> Hoe weet je de omvang van een matrix? Voorbeeld:<\/h2>\n<ul>\n<li> Bereken de omvang van de volgende matrix met afmeting 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\"> We zullen altijd beginnen met proberen te zien of de matrix de maximale rangorde heeft door de grootste ordedeterminant op te lossen. En als de determinant van deze orde gelijk is aan 0, zullen we doorgaan met het testen van determinanten van lagere orde totdat we er een vinden die anders is dan 0.<\/p>\n<p> In dit geval is het een matrix met afmeting 3\u00d74. <strong>Het zal daarom hoogstens rang 3 zijn<\/strong> , aangezien we geen enkele determinant van orde 4 kunnen maken. We nemen dus een willekeurige 3\u00d73-submatrix en kijken of de determinant ervan 0 is. We lossen bijvoorbeeld de determinant van de eerste 3 kolommen op met de Sarrus-regel:<\/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> De determinant van de kolommen 1, 2 en 3 is 0. We moeten nu een andere determinant proberen, bijvoorbeeld die van de kolommen 1, 2 en 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> Het leverde ons ook 0 op. We blijven daarom de determinanten van orde 3 testen om te zien of er andere zijn dan 0. We testen nu de determinant gevormd door de kolommen 1, 3 en 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> Probeer van de determinanten van orde 3 eenvoudigweg de determinant die bestaat uit de kolommen 2, 3 en 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> We hebben alle mogelijke 3&#215;3 determinanten van matrix A al geprobeerd, en aangezien geen van deze verschillend is van 0, <strong>heeft de matrix niet rang 3<\/strong> . Daarom zal het hoogstens rang 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-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> We zullen nu kijken of de matrix van rang 2 is. Om dit te doen, moeten we een vierkante submatrix van orde 2 vinden waarvan de determinant anders is dan 0. We zullen de 2\u00d72 submatrix in de linkerbovenhoek proberen:<\/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> We hebben een determinant van orde 2 gevonden die verschilt van 0 binnen de matrix. Bijgevolg <strong>is de matrix van rang 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\"> Problemen met de matrixscope opgelost<\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Bepaal de rangorde van de volgende 2\u00d72-matrix: <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We berekenen eerst de determinant van de gehele matrix:<\/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\"> We hebben een determinant van orde 2 gevonden die verschilt van 0. Daarom <strong>is de matrix van rang 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\">Oefening 2<\/h3>\n<p> Zoek de omvang van de volgende matrix met dimensie 2 \u00d7 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-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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst lossen we de determinant van de gehele matrix 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-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\"> De enige mogelijke 2\u00d72 determinant geeft 0, dus de matrix heeft niet rang 2.<\/p>\n<p class=\"has-text-align-left\"> Maar binnen de matrix zijn er andere 1&#215;1 determinanten dan 0, bijvoorbeeld:<\/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>De matrix is dus van rang 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\"> Oefening 3<\/h3>\n<p> Wat is de omvang van de volgende 3&#215;3 vierkante matrix? <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst wordt de determinant van de gehele matrix berekend met de Sarrus-regel:<\/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\"> De enige mogelijke 3\u00d73 determinant geeft 0, dus de matrix is niet van rang 3.<\/p>\n<p class=\"has-text-align-left\"> Maar binnen de matrix zijn er andere determinanten van orde 2 dan 0, bijvoorbeeld:<\/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\"> Daarom <strong>is de matrix van rang 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\">Oefening 4<\/h3>\n<p> Bereken de rangorde van de volgende matrix van orde 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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst wordt de determinant van de gehele matrix opgelost met de Sarrus-regel:<\/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\"> De determinant van de gehele matrix evalueert naar iets anders dan 0. Daarom heeft de matrix de maximale rang, dat wil zeggen <strong>rang 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\">Oefening 5<\/h3>\n<p> Wat is de rangorde van de volgende matrix van orde 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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Eerst wordt de determinant van de gehele matrix berekend met de Sarrus-regel:<\/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\"> De enige mogelijke 3\u00d73 determinant geeft 0, dus de matrix is niet van rang 3.<\/p>\n<p class=\"has-text-align-left\"> Maar binnen de matrix zijn er andere 2 \u00d7 2 determinanten dan 0, zoals:<\/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>De matrix is dus van rang 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\"> Oefening 6<\/h3>\n<p> Zoek de omvang van de volgende 3&#215;4-matrix: <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De matrix kan niet van rang 4 zijn, omdat we geen 4\u00d74 determinanten kunnen maken. Laten we dus kijken of het van rang 3 is door 3\u00d73 determinanten te berekenen:<\/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\"> De determinant van de eerste 3 kolommen geeft 0. De determinant van de laatste 3 kolommen geeft echter iets anders dan 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\"> Dus aangezien er binnenin een submatrix van orde 3 is waarvan de determinant verschilt van 0, <strong>is de matrix van rang 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\">Oefening 7<\/h3>\n<p> Bereken het bereik van de volgende 4&#215;3-matrix: <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De matrix kan niet van rang 4 zijn, omdat we geen enkele 4\u00d74-determinant kunnen oplossen. Laten we dus kijken of het van rang 3 is door alle mogelijke 3&#215;3 determinanten uit te voeren: <\/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\"> Omdat alle mogelijke 3\u00d73 determinanten 0 opleveren, heeft de matrix ook niet rang 3. We proberen nu de 2\u00d72 determinanten:<\/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\"> Omdat er binnen de matrix A een submatrix van orde 2 bestaat waarvan de determinant verschilt van 0, <strong>is de matrix van rang 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\">Oefening 8<\/h3>\n<p> Zoek het bereik van de volgende 4 \u00d7 4-matrix: <\/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>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> We moeten de determinant van de hele matrix oplossen om te zien of deze van rang 4 is.<\/p>\n<p class=\"has-text-align-left\"> En om de 4&#215;4-determinant op te lossen, moet je eerst bewerkingen uitvoeren met de rijen om op \u00e9\u00e9n na alle elementen in een kolom in nul te transformeren:<\/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\"> We berekenen nu de determinant door deputaten:<\/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\"> We vereenvoudigen de termen: <\/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\"> We berekenen de adjunct van 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\"> En tenslotte berekenen we de 3\u00d73 determinant met de Sarrusregel en de rekenmachine: <\/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\"> De 4&#215;4-determinant van de hele matrix geeft 0, dus matrix A zal niet van rang 4 zijn. Laten we nu eens kijken of er een andere 3&#215;3-determinant in zit dan 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-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>Matrix A is daarom van rang 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\"> Eigenschappen matrixbereik<\/h2>\n<ul>\n<li> Het bereik wordt niet gewijzigd als we een rij verwijderen die gevuld is met nullen, een kolom of een rij gevuld met 0.<\/li>\n<\/ul>\n<ul>\n<li> Het bereik van een matrix verandert niet als we de volgorde van twee parallelle rijen veranderen, of het nu rijen of kolommen zijn.<\/li>\n<\/ul>\n<ul>\n<li> De rangorde van een matrix is dezelfde als die van zijn transpositie.<\/li>\n<\/ul>\n<ul>\n<li> Als u een rij of kolom vermenigvuldigt met een ander getal dan 0, verandert de rangorde van de matrix niet.<\/li>\n<\/ul>\n<ul>\n<li> Het bereik van een tint verandert niet wanneer we een lijn (rij of kolom) elimineren die een lineaire combinatie is van andere lijnen die er evenwijdig aan lopen.<\/li>\n<\/ul>\n<ul>\n<li> Het bereik van een matrix verandert niet als we andere rijen optellen parallel aan een van de rijen (rijen of kolommen), vermenigvuldigd met een willekeurig getal. Dit is de reden waarom de rangorde van een matrix ook kan worden berekend met de Gaussiaanse methode. <\/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>Op deze pagina ziet u wat het is en hoe u het bereik van een matrix aan de hand van determinanten kunt berekenen. Daarnaast vind je voorbeelden en opgeloste oefeningen om te leren hoe je eenvoudig de omvang van een matrix kunt vinden. Bovendien ziet u ook de bereikeigenschappen van een matrix. Wat is de &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/rang-van-een-matrix\/\"> <span class=\"screen-reader-text\">Bereken de rangorde van een matrix op basis van determinanten<\/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":[39],"tags":[],"class_list":["post-338","post","type-post","status-publish","format-standard","hentry","category-determinant-van-een-matrix"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Bereken de rangorde van een matrix op basis van determinanten - 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\/nl\/rang-van-een-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bereken de rangorde van een matrix op basis van determinanten - Mathority\" \/>\n<meta property=\"og:description\" content=\"Op deze pagina ziet u wat het is en hoe u het bereik van een matrix aan de hand van determinanten kunt berekenen. 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