{"id":294,"date":"2023-07-06T17:31:42","date_gmt":"2023-07-06T17:31:42","guid":{"rendered":"https:\/\/mathority.org\/id\/peringkat-suatu-matriks\/"},"modified":"2023-07-06T17:31:42","modified_gmt":"2023-07-06T17:31:42","slug":"peringkat-suatu-matriks","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/peringkat-suatu-matriks\/","title":{"rendered":"Hitung pangkat suatu matriks berdasarkan determinannya"},"content":{"rendered":"<p>Di halaman ini Anda akan melihat apa itu dan bagaimana menghitung <strong>rentang suatu matriks<\/strong> berdasarkan determinan. Selain itu, Anda akan menemukan contoh dan latihan yang diselesaikan untuk mempelajari cara mencari luas suatu matriks dengan mudah. Selain itu, Anda juga akan melihat properti rentang suatu matriks.<\/p>\n<h2 class=\"wp-block-heading\"> Berapakah rank suatu matriks?<\/h2>\n<p> Definisi jangkauan suatu matriks adalah:<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> <strong>Pangkat suatu matriks<\/strong> adalah ordo submatriks persegi terbesar yang determinannya berbeda dengan 0.<\/p>\n<p> Pada halaman ini kita akan mempelajari tentang jangkauan suatu matriks dengan metode determinan, namun jangkauan suatu matriks juga dapat ditentukan dengan metode Gaussian, meskipun lebih lambat dan rumit.<\/p>\n<p> Setelah kita mengetahui rentang suatu matriks, kita akan melihat cara mencari rentang suatu matriks berdasarkan determinannya. Namun perlu diingat bahwa untuk menyelesaikan luas suatu matriks, Anda harus mengetahui cara menghitung<a href=\"https:\/\/mathority.org\/id\/contoh-aturan-sarrus-determinan-3x3-dan-latihan-penyelesaiannya\/\">determinan 3&#215;3<\/a> terlebih dahulu.<\/p>\n<h2 class=\"wp-block-heading\"> Bagaimana cara mengetahui luas suatu matriks? Contoh:<\/h2>\n<ul>\n<li> Hitung luas matriks berdimensi 3\u00d74 berikut: <\/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\"> Kita akan selalu memulai dengan mencoba melihat apakah matriks mempunyai rank maksimum dengan menyelesaikan determinan orde terbesar. Dan, jika determinan orde ini sama dengan 0, kita akan terus menguji determinan orde rendah hingga kita menemukan determinan selain 0.<\/p>\n<p> Dalam hal ini adalah matriks berdimensi 3\u00d74. <strong>Oleh karena itu, paling banyak akan berada pada peringkat 3<\/strong> , karena kita tidak dapat membuat determinan orde 4. Jadi kita ambil submatriks 3\u00d73 apa saja dan kita lihat apakah determinannya adalah 0. Misalnya, kita selesaikan determinan dari 3 kolom pertama dengan aturan Sarrus:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-819aaaa272025ce70b7852d00680483d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left( \\begin{tabular}{cccc}\\cellcolor[HTML]{ABEBC6}1 &amp; \\cellcolor[HTML]{ABEBC6}3 &amp; \\cellcolor[HTML]{ABEBC6}4  &amp; -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\\\[-2ex] \\cellcolor[HTML]{ABEBC6}0 &amp; \\cellcolor[HTML]{ABEBC6}2 &amp; \\cellcolor[HTML]{ABEBC6} 1 &amp; -1 \\\\ \\cellcolor[HTML]{ABEBC6} &amp; \\cellcolor[HTML]{ABEBC6} &amp;\\cellcolor[HTML]{ABEBC6} &amp; \\\\[-2ex]\\cellcolor[HTML]{ABEBC6} 3 &amp;\\cellcolor[HTML]{ABEBC6} -1 &amp; \\cellcolor[HTML]{ABEBC6} 7 &amp; 2                    \\end{tabular} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"76\" width=\"570\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aedcd597b0cd9cd0ad11ab1d99bd0e5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 3 &amp; 4 \\\\[1.1ex] 0 &amp; 2 &amp; 1   \\\\[1.1ex] 3 &amp; -1 &amp; 7  \\end{vmatrix} = 14 + 9 + 0 - 24 + 1 - 0 = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"318\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Penentu kolom 1, 2 dan 3 adalah 0. Sekarang kita harus mencoba determinan lain, misalnya kolom 1, 2 dan 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> Ini juga memberi kita 0. Oleh karena itu, kita terus menguji determinan berorde 3 untuk melihat apakah ada selain 0. Sekarang kita menguji determinan yang dibentuk oleh kolom 1, 3 dan 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> Dari determinan orde 3, coba saja determinan yang terdiri dari kolom 2, 3 dan 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> Kita telah mencoba semua kemungkinan determinan 3&#215;3 dari matriks A, dan karena tidak ada satupun yang berbeda dari 0, <strong>maka matriks tersebut tidak berpangkat 3<\/strong> . Oleh karena itu, paling banyak akan menjadi peringkat 2.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-157fd11377c30ccf66e64960e295866b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A) < 3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Sekarang kita akan melihat apakah matriksnya berpangkat 2. Untuk melakukannya, kita harus mencari submatriks persegi berorde 2 yang determinannya berbeda dari 0. Kita akan mencoba submatriks 2\u00d72 di pojok kiri atas:<\/p>\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> Kami menemukan determinan orde 2 yang berbeda dari 0 di dalam matriks. Akibatnya, <strong>matriks tersebut memiliki peringkat 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\"> Memecahkan Masalah Lingkup Matriks<\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Tentukan rank matriks 2\u00d72 berikut: <\/p>\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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita menghitung determinan seluruh matriks:<\/p>\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\"> Kami menemukan determinan orde 2 yang berbeda dari 0. Oleh karena itu, <strong>matriks tersebut memiliki peringkat 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\">Latihan 2<\/h3>\n<p> Tentukan luas matriks berdimensi 2\u00d72 berikut: <\/p>\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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama, kita selesaikan determinan seluruh matriks:<\/p>\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\"> Satu-satunya determinan 2\u00d72 yang mungkin menghasilkan 0, sehingga matriksnya tidak berpangkat 2.<\/p>\n<p class=\"has-text-align-left\"> Namun di dalam matriks tersebut terdapat determinan 1&#215;1 selain 0, contoh:<\/p>\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>Oleh karena itu, matriks tersebut berada pada peringkat 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\"> Latihan 3<\/h3>\n<p> Berapa luas matriks persegi 3&#215;3 berikut? <\/p>\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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama, determinan seluruh matriks dihitung dengan aturan Sarrus:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a1bda19a46e006dfc43ade0e92f189e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix} 1 &amp; -3 &amp; 2 \\\\[1.1ex] 2 &amp; 1 &amp; 4 \\\\[1.1ex] 1 &amp; 4 &amp; 2 \\end{vmatrix} = 2-12+16-2-16+12 \\bm{=0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"380\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Satu-satunya determinan 3\u00d73 yang mungkin menghasilkan 0, sehingga matriksnya tidak berpangkat 3.<\/p>\n<p class=\"has-text-align-left\"> Namun di dalam matriks tersebut terdapat determinan berorde 2 selain 0, misalnya:<\/p>\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\"> Oleh karena itu, <strong>matriks tersebut mempunyai rangking 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\">Latihan 4<\/h3>\n<p> Hitung rank matriks orde 3 berikut: <\/p>\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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama, determinan seluruh matriks diselesaikan dengan aturan Sarrus:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-819e9bea5c6d6d536a4dafba325ae45e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix} 3 &amp; -1 &amp; 1 \\\\[1.1ex] 4 &amp; -2 &amp; 3 \\\\[1.1ex] 2 &amp; 5 &amp; 2 \\end{vmatrix} = -12-6+20+4-45+8 =  -31\\bm{ \\neq0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"440\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Penentu seluruh matriks bernilai selain 0. Oleh karena itu, matriks tersebut mempunyai rangking maksimum, yaitu <strong>rangking 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\">Latihan 5<\/h3>\n<p> Berapa rank matriks orde 3 berikut ini? <\/p>\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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama, determinan seluruh matriks dihitung dengan aturan Sarrus:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e4eee911dcf234c3fa63177e533901af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix}2 &amp; 5 &amp; -1 \\\\[1.1ex] 3 &amp; -2 &amp; -4 \\\\[1.1ex] 5 &amp; 3 &amp; -5 \\end{vmatrix} =20-100-9-10+24+75 \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"411\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Satu-satunya determinan 3\u00d73 yang mungkin menghasilkan 0, sehingga matriksnya tidak berpangkat 3.<\/p>\n<p class=\"has-text-align-left\"> Namun di dalam matriks tersebut terdapat determinan 2\u00d72 selain 0, seperti:<\/p>\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>Oleh karena itu, matriksnya berada pada peringkat 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\"> Latihan 6<\/h3>\n<p> Tentukan luas matriks 3&#215;4 berikut: <\/p>\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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Matriksnya tidak boleh berpangkat 4, karena kita tidak bisa membuat determinan 4&#215;4. Jadi mari kita lihat apakah ia menduduki peringkat 3 dengan menghitung determinan 3\u00d73:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2897851f49a9556fc03aded5f1495297_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} A \\end{vmatrix}= \\begin{vmatrix}3 &amp; 2 &amp; -4  \\\\[1.1ex] 2 &amp; -2 &amp; -3  \\\\[1.1ex] 5 &amp; 0 &amp; -7 \\end{vmatrix} =42-30+0-40-0+28 \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"393\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Penentu 3 kolom pertama menghasilkan 0. Namun, determinan 3 kolom terakhir menghasilkan selain 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\"> Jadi, karena di dalamnya terdapat submatriks berorde 3 yang determinannya berbeda dengan 0, <strong>maka matriks tersebut berpangkat 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\">Latihan 7<\/h3>\n<p> Hitung rentang matriks 4&#215;3 berikut: <\/p>\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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Matriksnya tidak boleh mempunyai peringkat 4, karena kita tidak dapat menyelesaikan determinan 4&#215;4 apa pun. Jadi mari kita lihat apakah ia berada di peringkat 3 dengan melakukan semua kemungkinan determinan 3&#215;3: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1eec1befc1515b4405529ede01c55618_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 3 &amp; 4 &amp; -5 \\\\[1.1ex] 5 &amp; -2 &amp; -9\\end{vmatrix} \\bm{= 0} \\qquad \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 5 &amp; -2 &amp; -9 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"308\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fa1de344c0bb747c9861afd4de5fa7c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 &amp; -2 \\\\[1.1ex] 3 &amp; 4 &amp; -5 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0} \\qquad \\begin{vmatrix} 3 &amp; 4 &amp; -5 \\\\[1.1ex] 5 &amp; -2 &amp; -9 \\\\[1.1ex] -2 &amp; -7 &amp; 3\\end{vmatrix} \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"322\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena semua determinan 3\u00d73 yang mungkin menghasilkan 0, matriksnya juga tidak berperingkat 3. Kami sekarang mencoba determinan 2\u00d72:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-981d085760dd1b1dd46aab17f1d7ba78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; -3 \\\\[1.1ex] 3 &amp; 4  \\end{vmatrix} =13 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"127\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena di dalam matriks A terdapat submatriks berorde 2 yang determinannya berbeda dengan 0, <strong>maka matriks tersebut berpangkat 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\">Latihan 8<\/h3>\n<p> Tentukan range matriks 4\u00d74 berikut: <\/p>\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>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Kita harus menyelesaikan determinan seluruh matriks untuk melihat apakah matriks tersebut mempunyai rangking 4.<\/p>\n<p class=\"has-text-align-left\"> Dan untuk menyelesaikan determinan 4&#215;4, Anda harus terlebih dahulu melakukan operasi dengan baris untuk mengubah semua kecuali satu elemen dalam kolom menjadi nol:<\/p>\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\"> Kami sekarang menghitung determinan berdasarkan deputi:<\/p>\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\"> Kami menyederhanakan persyaratannya: <\/p>\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\"> Kami menghitung adjoin dari 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\"> Dan terakhir, kita menghitung determinan 3\u00d73 dengan aturan Sarrus dan kalkulator: <\/p>\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\"> Penentu 4&#215;4 dari keseluruhan matriks menghasilkan 0, sehingga matriks A tidak mempunyai rangking 4. Jadi sekarang mari kita lihat apakah matriks tersebut mempunyai determinan 3&#215;3 selain 0 di dalamnya:<\/p>\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>Oleh karena itu, matriks A berada pada peringkat 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\"> Properti Rentang Matriks<\/h2>\n<ul>\n<li> Rentang tersebut tidak diubah jika kita menghapus baris yang diisi dengan nol, baik kolom, atau baris yang diisi dengan 0.<\/li>\n<\/ul>\n<ul>\n<li> Kisaran suatu matriks tidak berubah jika kita mengubah urutan dua baris sejajar, baik baris maupun kolom.<\/li>\n<\/ul>\n<ul>\n<li> Pangkat suatu matriks sama dengan pangkat transposnya.<\/li>\n<\/ul>\n<ul>\n<li> Jika baris atau kolom dikalikan dengan angka selain 0, maka pangkat matriks tidak berubah.<\/li>\n<\/ul>\n<ul>\n<li> Kisaran rona tidak berubah ketika kita menghilangkan suatu garis (baris atau kolom) yang merupakan kombinasi linier dari garis-garis lain yang sejajar dengannya.<\/li>\n<\/ul>\n<ul>\n<li> Kisaran suatu matriks tidak berubah jika kita menjumlahkan baris-baris lain yang sejajar dengan salah satu baris (baris atau kolom) dikalikan dengan bilangan berapa pun. Inilah sebabnya mengapa rank suatu matriks juga dapat dihitung dengan metode Gaussian. <\/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>Di halaman ini Anda akan melihat apa itu dan bagaimana menghitung rentang suatu matriks berdasarkan determinan. Selain itu, Anda akan menemukan contoh dan latihan yang diselesaikan untuk mempelajari cara mencari luas suatu matriks dengan mudah. Selain itu, Anda juga akan melihat properti rentang suatu matriks. Berapakah rank suatu matriks? Definisi jangkauan suatu matriks adalah: Pangkat &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/peringkat-suatu-matriks\/\"> <span class=\"screen-reader-text\">Hitung pangkat suatu matriks berdasarkan determinannya<\/span> Selengkapnya &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-294","post","type-post","status-publish","format-standard","hentry","category-penentu-suatu-matriks"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Hitung pangkat suatu matriks berdasarkan determinan - 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\/id\/peringkat-suatu-matriks\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Hitung pangkat suatu matriks berdasarkan determinan - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan melihat apa itu dan bagaimana menghitung rentang suatu matriks berdasarkan determinan. Selain itu, Anda akan menemukan contoh dan latihan yang diselesaikan untuk mempelajari cara mencari luas suatu matriks dengan mudah. Selain itu, Anda juga akan melihat properti rentang suatu matriks. Berapakah rank suatu matriks? 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