{"id":287,"date":"2023-07-06T20:03:12","date_gmt":"2023-07-06T20:03:12","guid":{"rendered":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/"},"modified":"2023-07-06T20:03:12","modified_gmt":"2023-07-06T20:03:12","slug":"contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/","title":{"rendered":"Properti penentu"},"content":{"rendered":"<p>Pada bagian ini, kita akan melihat apa saja <strong>sifat-sifat determinan<\/strong> . Kami juga mendemonstrasikan setiap properti dengan sebuah contoh sehingga Anda memahaminya sepenuhnya. Dan, sebagai tambahan, Anda akan menemukan latihan yang berkaitan dengan sifat-sifat determinan.<\/p>\n<p> Di bawah ini kami akan menjelaskan masing-masing sifat determinan satu per satu, namun jika mau, Anda dapat langsung melompat ke <strong>tabel ringkasan<\/strong> di bawah ini. \ud83d\ude09<\/p>\n<h2 class=\"wp-block-heading\"> Properti 1: Penentu matriks yang ditransposisikan <\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\"> Penentu suatu matriks setara dengan determinan matriks yang ditransposisikan.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2f3228b0db84cb6cff32c1157107dfd7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert A \\rvert = \\lvert A^t \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7e39b73d52e436c660c2c9f2eeed39f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert A \\rvert =  \\begin{vmatrix} 2 &amp; 3 \\\\[1.1ex] 1 &amp; 5  \\end{vmatrix} =  2 \\cdot 5 - 1 \\cdot 3 = 10 - 3 = \\bm{7}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"301\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita transposisi matriks 2&#215;2 dan mencari determinannya. Perhatikan bahwa kita memperoleh hasil yang sama seperti sebelumnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fc16abe425fb139cb3a6b7ba7e3b1915_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert A^t \\rvert =  \\begin{vmatrix} 2 &amp; 1 \\\\[1.1ex] 3 &amp; 5  \\end{vmatrix} =  2 \\cdot 5 - 3 \\cdot 1 = 10 - 3 = \\bm{7}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"307\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Sifat 2: Penentu yang baris atau kolomnya diisi angka nol <\/h2>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\"> Jika suatu determinan memiliki baris atau kolom yang diisi dengan nol, maka determinannya menghasilkan 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-54df933d2167697d926c25dd9554d90a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} a_{11} &amp; 0 &amp; a_{13} \\\\[1.1ex] a_{21} &amp; 0 &amp; a_{23} \\\\[1.1ex] a_{31} &amp; 0 &amp; a_{33}\\end{vmatrix}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"132\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-95bc2d762871764f41176acc052a633c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 5 &amp; 6 &amp; 2 \\\\[1.1ex] 0 &amp; 0 &amp; 0 \\\\[1.1ex] -3 &amp; 1 &amp; 4 \\end{vmatrix}   =  \\bm{0} \\qquad \\qquad \\begin{vmatrix} 1 &amp; -5 &amp; 0 \\\\[1.1ex] 6 &amp; 2 &amp; 0 \\\\[1.1ex] 1 &amp; 3 &amp; 0 \\end{vmatrix} = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dalam kedua contoh ini, determinannya bernilai 0. Karena baris kedua dari determinan pertama semuanya nol dan kolom ketiga dari determinan kedua juga semuanya nol.<\/p>\n<h2 class=\"wp-block-heading\"> Sifat 3: Penentu dengan dua baris atau kolom yang sama <\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-110\"><\/div>\n<\/div>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Jika suatu determinan mempunyai dua baris atau dua kolom yang sama atau ganda, maka determinannya adalah nol (0).<\/p>\n<p> Oleh karena itu, jika terdapat kombinasi linier antara baris atau kolom, yaitu bergantung linier, maka determinannya menghasilkan 0.<\/p>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c31b59570d4f89e8c7e7aa9f922977c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 3 &amp; 4 &amp; 4 \\\\[1.1ex] -1 &amp; 5 &amp; 5 \\\\[1.1ex] 6 &amp; 2 &amp; 2 \\end{vmatrix} =  0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"117\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dalam hal ini determinannya menghasilkan 0 karena kolom 2 dan 3 sama.<\/p>\n<h2 class=\"wp-block-heading\"> Properti 4: Ubah baris atau kolom determinan <\/h2>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\"> Jika dua baris atau dua kolom dimodifikasi relatif satu sama lain, determinannya memberikan hasil yang sama tetapi dengan tanda yang berbeda.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-de79fc53e94c9a30d8a271d42d4e3494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} a &amp; b &amp; c \\\\[1.1ex] d &amp; e &amp; f \\\\[1.1ex] g &amp; h &amp; i \\end{vmatrix}= - \\begin{vmatrix} a &amp; c &amp; b \\\\[1.1ex] d &amp; f &amp; e \\\\[1.1ex] g &amp; i &amp; h \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"185\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c2e4806318fa67998b339383a9dc9ea5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 3 &amp; 2 &amp; -4 \\\\[1.1ex] 1 &amp; 5 &amp; 6 \\\\[1.1ex] 1 &amp; 0 &amp; -3 \\end{vmatrix} = \\displaystyle -45 +12+0+20-0+6=  \\bm{-7}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"357\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita mengubah urutan kolom 2 dan 3 relatif satu sama lain. Perhatikan bahwa hasilnya sama tetapi dengan tanda yang berbeda:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4a4de8b8cf37df2c3cce69d16a19a578_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 3 &amp; -4 &amp; 2 \\\\[1.1ex] 1 &amp; 6 &amp; 5 \\\\[1.1ex] 1 &amp; -3 &amp; 0 \\end{vmatrix}   = \\displaystyle 0-20-6-12+45-0=  \\bm{+7}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"343\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Properti 5: Kalikan baris determinan dengan skalar <\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\"> Mengalikan seluruh elemen pada suatu baris atau kolom dengan bilangan real sama dengan mengalikan hasil determinan dengan bilangan tersebut. <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-06cf5f62a3d703b43bb68b319839df26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} k \\cdot a_{11} &amp;  k \\cdot a_{12} &amp; k \\cdot a_{13} \\\\[1.1ex] a_{21} &amp;  a_{22} &amp; a_{23} \\\\[1.1ex] a_{31} &amp;  a_{32} &amp; a_{33} \\end{vmatrix} =k \\cdot \\begin{vmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\[1.1ex] a_{21} &amp;  a_{22} &amp; a_{23} \\\\[1.1ex] a_{31} &amp;  a_{32} &amp; a_{33} \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"342\" 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-a7b38fe06dab0bbdbfef384b3e403fed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} k \\cdot a_{11} &amp; a_{12} &amp; a_{13} \\\\[1.1ex] k \\cdot a_{21} &amp;  a_{22} &amp; a_{23} \\\\[1.1ex] k \\cdot a_{31} &amp;  a_{32} &amp; a_{33} \\end{vmatrix} =k \\cdot \\begin{vmatrix} a_{11} &amp; a_{12} &amp; a_{13} \\\\[1.1ex] a_{21} &amp;  a_{22} &amp; a_{23} \\\\[1.1ex] a_{31} &amp;  a_{32} &amp; a_{33} \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"297\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<\/div>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-92b404bb7ad8bbdd59c8c54c1619c37d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle   \\begin{vmatrix} 2 &amp; 3 \\\\[1.1ex] 1 &amp; 4  \\end{vmatrix}   = 8-3= \\bm{5}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"138\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita ambil determinan yang sama dan mengalikan seluruh garis dengan 2. Anda akan melihat bahwa hasilnya sama dengan determinan sebelumnya tetapi dikalikan dengan 2, atau 10:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-57ae1ba33c0d108f08ac9d0b5cb4a81b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 2 \\cdot 2 &amp; 2 \\cdot 3 \\\\[1.1ex] 1 &amp; 4  \\end{vmatrix}   =  \\begin{vmatrix} 4 &amp; 6 \\\\[1.1ex] 1 &amp; 4  \\end{vmatrix} = 16-6 =\\bm{10}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"270\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Sifat 6: Penentu hasil kali matriks <\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-111\"><\/div>\n<\/div>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\"> Penentu hasil kali dua matriks sama dengan hasil kali determinan masing-masing matriks secara terpisah.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-066ee1431d90a4c2cb6febe8a381cc69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\lvert A \\cdot B \\rvert = \\lvert A \\rvert \\cdot \\lvert B \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<p> Untuk mendemonstrasikan sifat determinan ini, kita akan menghitung determinan perkalian dua matriks berikut dengan dua cara yang mungkin:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-38de0ca99ad15f40bd94f653cffacf8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} 1 &amp; 3 \\\\[1.1ex] 2 &amp; 5 \\end{pmatrix}\\quad B=\\begin{pmatrix} 4 &amp; 2 \\\\[1.1ex] 1 &amp; -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"229\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Pertama-tama kita kalikan kedua matriks tersebut, lalu kita hitung determinan matriks yang dihasilkan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2ad18d1637b581038b7866030d6ac9a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\left| A \\cdot B \\right| =\\left| \\begin{pmatrix} 1 &amp; 3 \\\\[1.1ex] 2 &amp; 5  \\end{pmatrix} \\cdot \\begin{pmatrix} 4 &amp; 2 \\\\[1.1ex] 1 &amp; -1  \\end{pmatrix}\\right|  = \\left| \\begin{pmatrix} 7 &amp; -1 \\\\[1.1ex] 13 &amp; -1  \\end{pmatrix} \\right|  = -7 - (-13) = \\bm{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"500\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita menghitung determinan masing-masing matriks secara terpisah dan kemudian mengalikan hasilnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5f7c97d5a832d3985bf1d5e9d4d44401_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert A \\rvert \\cdot \\lvert B \\rvert =  \\left| \\begin{pmatrix} 1 &amp; 3 \\\\[1.1ex] 2 &amp; 5  \\end{pmatrix} \\right| \\cdot \\left| \\begin{pmatrix} 4 &amp; 2 \\\\[1.1ex] 1 &amp; -1  \\end{pmatrix}\\right| = -1\\cdot (-6)= \\bm{6}\" title=\"Rendered by QuickLaTeX.com\" height=\"55\" width=\"384\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Seperti yang Anda lihat, mengerjakan perkalian matriks terlebih dahulu lalu determinannya memberikan hasil yang sama seperti mengerjakan determinan setiap matriks terlebih dahulu lalu mengalikan hasilnya.<\/p>\n<p> Sebaliknya, syarat ini tidak berlaku pada operasi penjumlahan dan pengurangan, artinya determinan penjumlahan (atau pengurangan) dua matriks tidak memberikan hasil yang sama dengan penjumlahan (atau pengurangan) determinan matriks. dua matriks secara terpisah. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Properti 7: Penentu matriks invers <\/h2>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p align=\"LEFT\"> Jika suatu matriks dapat dibalik, maka determinan inversnya sama dengan invers determinan matriks aslinya.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-460186cec7a5d86981bd5a14e3b1dcf0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} A^{-1} \\end{vmatrix} = \\cfrac{1}{\\lvert A \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"91\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<\/div>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<p> Kita akan memverifikasi sifat ini dengan terlebih dahulu menghitung invers suatu matriks dan kemudian menyelesaikan determinannya. Kita akan melihat bahwa hasilnya setara dengan mencari determinan matriks asli dan membalikkannya.<\/p>\n<p> Oleh karena itu, kami membalikkan matriks berikut dan menghitung determinannya: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2c77c10006d35ebc5273553fb84356e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} 4 &amp; 2 \\\\[1.1ex] 7 &amp; 4 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" 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-8a9315a8add365cd5f077c52476a827d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A^{-1}= \\begin{pmatrix} 2 &amp; -1 \\\\[1.1ex] -\\frac{7}{2} &amp; 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"143\" 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-a10c873ff6c101cd2b239388393c268b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}A^{-1} \\end{vmatrix}= \\begin{vmatrix} 2 &amp; -1 \\\\[1.1ex] -\\frac{7}{2} &amp; 2 \\end{vmatrix} = 4-\\cfrac{7}{2} =\\cfrac{8}{2}-\\cfrac{7}{2} = \\cfrac{\\bm{1}}{\\bm{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"307\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dan sekarang kita selesaikan determinan matriks asli dan lakukan inversnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-850e5404f5352782327918caab3e1440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}A\\end{vmatrix}= \\begin{pmatrix} 4 &amp; 2 \\\\[1.1ex] 7 &amp; 4 \\end{pmatrix}=16-14=2\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"221\" 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-427c70eab8ecba40b6dcde2a6e03abd2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}A^{-1}\\end{vmatrix}= \\cfrac{\\bm{1}}{\\bm{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Seperti yang Anda lihat, hasil dari kedua operasi tersebut identik. Oleh karena itu, properti tersebut terbukti.<\/p>\n<h2 class=\"wp-block-heading\"> Properti 8: Gantikan garis penentu<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Barisan suatu determinan dapat diganti dengan menjumlahkan (atau mengurangkan) baris yang sama ditambah (atau dikurangi) baris lain dikalikan dengan suatu bilangan.<\/p>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<p> Dengan contoh berikut kita akan memeriksa properti ini. Kita hitung dulu determinannya, lalu kita operasikan baris determinannya dan hitung ulang hasilnya. Anda akan melihat bagaimana kami memperoleh hasil yang sama dalam kedua kasus.<\/p>\n<p> Jadi, mari kita hitung dulu determinan 3&#215;3 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-4ccd76fc3a2b7cd7afc7d8f9de8ffde1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 2 &amp; 1 &amp; -1 \\\\[1.1ex] 3 &amp; 0 &amp; 1 \\\\[1.1ex] 0 &amp; -3 &amp; 6 \\end{vmatrix} \\displaystyle=0+0+9-0+6-18 =  \\bm{-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"338\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang, <strong>pada baris 2, kita tambahkan baris pertama dikalikan 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-4bc2e6bd78446fb68f29b4a5503a6828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; 1 &amp; -1 \\\\[1.1ex] 3 &amp; 0 &amp; 1 \\\\[1.1ex] 0 &amp; -3 &amp; 6 \\end{vmatrix} \\begin{matrix} \\\\[1.1ex] \\xrightarrow{f_2 + 2f_1}  \\\\[1.1ex] \\  \\end{matrix} \\begin{vmatrix} 2 &amp; 1 &amp; -1 \\\\[1.1ex] 7 &amp; 2 &amp; -1 \\\\[1.1ex] 0 &amp; -3 &amp; 6 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"254\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dan kita menyelesaikan determinannya setelah mengubah salah satu garisnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-045eb6f32420fbbf538a9e0a540ce119_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2 &amp; 1 &amp; -1 \\\\[1.1ex] 7 &amp; 2 &amp; -1 \\\\[1.1ex] 0 &amp; -3 &amp; 6 \\end{vmatrix} = 24+0+21-0-6-42=\\bm{-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"355\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam kedua kasus tersebut, hasilnya adalah -3. Jadi, terlihat bahwa hasil suatu determinan tidak berubah jika suatu baris diganti dengan jumlah baris yang sama ditambah baris lainnya dikalikan suatu bilangan. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Properti 9: Penentu matriks segitiga<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Penentu matriks segitiga adalah hasil kali elemen-elemen diagonal utamanya.<\/p>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<p> Kita akan menyelesaikan determinan matriks segitiga berikut sebagai 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-40ebfa5f9f06e63ad1325d9331a57bde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} 2 &amp; 3 &amp; 5 \\\\[1.1ex] 0 &amp; -1 &amp; 7 \\\\[1.1ex] 0 &amp; 0 &amp; 4 \\end{vmatrix} \\displaystyle= 2 \\cdot (-1) \\cdot 4 =  \\bm{-8}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"234\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Sifat 10: Penentu matriks diagonal<\/h2>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Penentu suatu matriks diagonal sama dengan perkalian elemen-elemen diagonal utamanya.<\/p>\n<h3 style=\"color:#00B0FF\"> Contoh:<\/h3>\n<p> Mari kita ambil contoh determinan matriks diagonal 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-56e1b3093685a1af729310752b03dfc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix}5 &amp; 0 &amp; 0 \\\\[1.1ex] 0 &amp; 3 &amp; 0 \\\\[1.1ex] 0 &amp; 0 &amp; -2 \\end{vmatrix} \\displaystyle= 5 \\cdot 3 \\cdot (-2) =  \\bm{-30}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"243\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Tabel ringkasan sifat-sifat determinan <\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-116\"><\/div>\n<\/div>\n<p> Sifat-sifat determinan yang dijelaskan dapat dirangkum dalam tabel berikut: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/proprietes-des-determinants.webp\" alt=\"sifat determinan\" class=\"wp-image-3447\" width=\"774\" height=\"669\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"> Latihan soal dengan sifat-sifat determinan<\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Selesaikan determinan 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-97dfc1ebfc5db73750870911108bd447_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 3 &amp; 1 &amp; 0 \\\\[1.1ex] 4 &amp; 2 &amp; 0 \\\\[1.1ex] -1 &amp; 6 &amp; 0 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"81\" 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\"> Jika suatu determinan memiliki baris atau kolom yang diisi dengan nol, determinan tersebut mengembalikan 0 (properti 2). Jadi <strong>hasil determinannya adalah 0, karena kolom ketiga diisi angka nol.<\/strong><\/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> Selesaikan determinan 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-bdb315ba588fe5fdfb03c7fea2857b16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 4 &amp; 2 &amp; -3 &amp; 5 \\\\[1.1ex] 1 &amp; 5 &amp; 3 &amp; 2 \\\\[1.1ex]4 &amp; 2 &amp; -3 &amp; 5 \\\\[1.1ex] -2 &amp; 0 &amp; 4 &amp; 3 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"119\" 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\"> Jika determinan memiliki dua baris atau dua kolom yang sama atau ganda, determinan mengembalikan 0 (properti 3). Jadi <strong>hasil determinannya adalah 0, karena baris pertama dan baris ketiga sama.<\/strong><\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Latihan 3<\/h3>\n<p> Hitung determinan 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-96c5cfee4c4189e49b54fdf43b2a0457_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 1 &amp; 0 &amp; 2 &amp; 2 \\\\[1.1ex] 3 &amp; 1 &amp; 5 &amp; 6 \\\\[1.1ex] 1 &amp; 3 &amp; -2 &amp; 2 \\\\[1.1ex] 2 &amp; 2 &amp; 0 &amp; 4 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"106\" 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\"> Jika determinan memiliki dua baris atau dua kolom yang sama atau ganda, determinan mengembalikan 0 (properti 3). Jadi <strong>hasil determinannya adalah 0, karena kolom keempat adalah dua kali kolom pertama.<\/strong> <\/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 4<\/h3>\n<p> Kita mengetahui hasil suatu determinan, meskipun kita tidak mengetahui elemen-elemen matriksnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e20d9016edd52f18d3ffc928d2658efe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} a &amp; b \\\\[1.1ex] c &amp; d  \\end{vmatrix} = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"77\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dari hasil determinan sebelumnya dan sifat-sifat determinannya, hitunglah hasil determinan 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-9c20de8d82171dc8fb784e2549521f37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\mathbf{a} \\bm{)} \\ \\begin{vmatrix} a &amp; c  \\\\[1.1ex] b &amp; d  \\end{vmatrix} \\qquad \\mathbf{b} \\bm{)} \\ \\begin{vmatrix} b &amp; a  \\\\[1.1ex] d &amp; c  \\end{vmatrix} \\qquad \\mathbf{c} \\bm{)} \\ \\begin{vmatrix} a &amp; 3b  \\\\[1.1ex] c &amp; 3d  \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"301\" 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\"> <strong>Untuk)<\/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-6ee3d744a077ee8fdc07e806f13286be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} a &amp; c  \\\\ b &amp; d  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"55\" style=\"vertical-align: -17px;\"><\/p>\n<p> adalah matriks yang ditransposisikan dari<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1565c44a2743bb11e27ba41203073382_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} a &amp; b  \\\\ c &amp; d  \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"55\" style=\"vertical-align: -17px;\"><\/p>\n<p> . Dan determinan suatu matriks sama dengan determinan matriks yang ditransposisikan (properti 1). Jadi, <strong>hasil determinannya juga 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-5bf7ae0a2ff32d75a6f7abafb623639c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} a &amp; c  \\\\[1.1ex] b &amp; d  \\end{vmatrix}=\\begin{vmatrix} a &amp; b \\\\[1.1ex] c &amp; d  \\end{vmatrix}=\\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"147\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>b)<\/strong> Dalam menentukan<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9255a916d06e3d7689e830d0456f5c74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} b &amp; a  \\\\ d &amp; c  \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"43\" style=\"vertical-align: -17px;\"><\/p>\n<p> kolom 1 dan 2 telah dimodifikasi sehubungan dengan penentu pernyataan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3e84a753ce5d5bfe9dd6831b42857b30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} a &amp; b \\\\ c &amp; d  \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"43\" style=\"vertical-align: -17px;\"><\/p>\n<p> . Oleh karena itu, menurut sifat 4, <strong>hasilnya sama dengan hasil penentu pernyataan tersebut tetapi dengan tanda yang berbeda, yaitu -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-d6d14aa2f6b8c7d1fd064daef8dd0eec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} b &amp; a  \\\\[1.1ex] d &amp; c  \\end{vmatrix} = - \\begin{vmatrix} a &amp; b \\\\[1.1ex] c &amp; d  \\end{vmatrix}= \\bm{-3}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"178\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>c)<\/strong> Dalam menentukan<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f343927516e13005f5d744228bfdfec6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{vmatrix} a &amp; 3b  \\\\ c &amp; 3d  \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"52\" style=\"vertical-align: -17px;\"><\/p>\n<p> seluruh kolom kedua determinan pernyataan tersebut telah dikalikan 3. Oleh karena itu, dari sifat 5 dapat disimpulkan bahwa <strong>hasilnya juga merupakan hasil determinan pernyataan tersebut dikalikan 3, yaitu 9.<\/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-b1a7ffec429367a2fd967a197d0299d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} a &amp; 3b  \\\\[1.1ex] c &amp; 3d  \\end{vmatrix} =3 \\begin{vmatrix} a &amp; b \\\\[1.1ex] c &amp; d  \\end{vmatrix} =3 \\cdot 3 = \\bm{9}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"222\" 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\">Latihan 5<\/h3>\n<p> Kita mengetahui hasil dari dua determinan 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-e938c40ce401263da9835fa77fc9a1dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\vert A \\vert = \\begin{vmatrix} 1 &amp; 2 &amp; 0 &amp; 1 \\\\[1.1ex] -2 &amp; -1 &amp; 1 &amp; 0 \\\\[1.1ex] 1 &amp; 3 &amp; 3 &amp; -1 \\\\[1.1ex] 3 &amp; 4 &amp; 1 &amp; 1 \\end{vmatrix}=8\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"215\" 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-6cea0d73e66099f2a10f71f7267baee9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\vert B \\vert = \\begin{vmatrix} 0 &amp; 1 &amp; 3 &amp; 2 \\\\[1.1ex] -1 &amp; -2 &amp; 0 &amp; 0 \\\\[1.1ex] 3 &amp; 1 &amp; 1 &amp; 2 \\\\[1.1ex] -1 &amp; 2 &amp; 3 &amp; 1 \\end{vmatrix} = - 4\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dari informasi tersebut, hitunglah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cce5c21696d6cc754d3b49cb7ea5457b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\vert A \\cdot B \\vert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"46\" style=\"vertical-align: -5px;\"><\/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\"> Untuk menghitung hasil determinan tidak perlu mengalikan matriks 4\u00d74. Karena <strong>determinan hasil kali dua matriks sama dengan hasil kali determinan masing-masing matriks secara terpisah<\/strong> (sifat 6). Belum: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b1e9aef3e2499e7ed6d085319ce955e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vert A \\cdot B \\vert  = \\vert A \\vert \\cdot \\vert B \\vert = 8 \\cdot (-4) = \\bm{-32}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"268\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Pada bagian ini, kita akan melihat apa saja sifat-sifat determinan . Kami juga mendemonstrasikan setiap properti dengan sebuah contoh sehingga Anda memahaminya sepenuhnya. Dan, sebagai tambahan, Anda akan menemukan latihan yang berkaitan dengan sifat-sifat determinan. Di bawah ini kami akan menjelaskan masing-masing sifat determinan satu per satu, namun jika mau, Anda dapat langsung melompat ke &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/\"> <span class=\"screen-reader-text\">Properti penentu<\/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-287","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>Sifat-sifat 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\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Sifat-sifat determinan - Mathority\" \/>\n<meta property=\"og:description\" content=\"Pada bagian ini, kita akan melihat apa saja sifat-sifat determinan . Kami juga mendemonstrasikan setiap properti dengan sebuah contoh sehingga Anda memahaminya sepenuhnya. Dan, sebagai tambahan, Anda akan menemukan latihan yang berkaitan dengan sifat-sifat determinan. Di bawah ini kami akan menjelaskan masing-masing sifat determinan satu per satu, namun jika mau, Anda dapat langsung melompat ke &hellip; Properti penentu Selengkapnya &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T20:03:12+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2f3228b0db84cb6cff32c1157107dfd7_l3.png\" \/>\n<meta name=\"author\" content=\"Tim Mathority\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Ditulis oleh\" \/>\n\t<meta name=\"twitter:data1\" content=\"Tim Mathority\" \/>\n\t<meta name=\"twitter:label2\" content=\"Estimasi waktu membaca\" \/>\n\t<meta name=\"twitter:data2\" content=\"5 menit\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Properti penentu\",\"datePublished\":\"2023-07-06T20:03:12+00:00\",\"dateModified\":\"2023-07-06T20:03:12+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/\"},\"wordCount\":913,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/id\/#organization\"},\"articleSection\":[\"Penentu suatu matriks\"],\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/\",\"url\":\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/\",\"name\":\"Sifat-sifat determinan - Mathority\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/#website\"},\"datePublished\":\"2023-07-06T20:03:12+00:00\",\"dateModified\":\"2023-07-06T20:03:12+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/#breadcrumb\"},\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/id\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Properti penentu\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/id\/#website\",\"url\":\"https:\/\/mathority.org\/id\/\",\"name\":\"Mathority\",\"description\":\"Di mana rasa ingin tahu bertemu dengan perhitungan!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/id\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/id\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"id\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/id\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/id\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"id\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/id\/wp-content\/uploads\/2023\/09\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/id\/wp-content\/uploads\/2023\/09\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\",\"name\":\"Tim Mathority\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"id\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Tim Mathority\"},\"sameAs\":[\"http:\/\/mathority.org\/id\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Sifat-sifat determinan - Mathority","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/","og_locale":"id_ID","og_type":"article","og_title":"Sifat-sifat determinan - Mathority","og_description":"Pada bagian ini, kita akan melihat apa saja sifat-sifat determinan . Kami juga mendemonstrasikan setiap properti dengan sebuah contoh sehingga Anda memahaminya sepenuhnya. Dan, sebagai tambahan, Anda akan menemukan latihan yang berkaitan dengan sifat-sifat determinan. Di bawah ini kami akan menjelaskan masing-masing sifat determinan satu per satu, namun jika mau, Anda dapat langsung melompat ke &hellip; Properti penentu Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/","article_published_time":"2023-07-06T20:03:12+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2f3228b0db84cb6cff32c1157107dfd7_l3.png"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"5 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Properti penentu","datePublished":"2023-07-06T20:03:12+00:00","dateModified":"2023-07-06T20:03:12+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/"},"wordCount":913,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"articleSection":["Penentu suatu matriks"],"inLanguage":"id","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/","url":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/","name":"Sifat-sifat determinan - Mathority","isPartOf":{"@id":"https:\/\/mathority.org\/id\/#website"},"datePublished":"2023-07-06T20:03:12+00:00","dateModified":"2023-07-06T20:03:12+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/#breadcrumb"},"inLanguage":"id","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/id\/contoh-sifat-sifat-determinan-dan-latihan-2x2-3x3\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/id\/"},{"@type":"ListItem","position":2,"name":"Properti penentu"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/id\/#website","url":"https:\/\/mathority.org\/id\/","name":"Mathority","description":"Di mana rasa ingin tahu bertemu dengan perhitungan!","publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/id\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"id"},{"@type":"Organization","@id":"https:\/\/mathority.org\/id\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/id\/","logo":{"@type":"ImageObject","inLanguage":"id","@id":"https:\/\/mathority.org\/id\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/id\/wp-content\/uploads\/2023\/09\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/id\/wp-content\/uploads\/2023\/09\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/id\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38","name":"Tim Mathority","image":{"@type":"ImageObject","inLanguage":"id","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Tim Mathority"},"sameAs":["http:\/\/mathority.org\/id"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts\/287","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/comments?post=287"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts\/287\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/media?parent=287"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/categories?post=287"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/tags?post=287"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}