{"id":305,"date":"2023-07-06T14:13:39","date_gmt":"2023-07-06T14:13:39","guid":{"rendered":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/"},"modified":"2023-07-06T14:13:39","modified_gmt":"2023-07-06T14:13:39","slug":"contoh-aturan-dan-latihan-cramer-yang-diselesaikan","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/","title":{"rendered":"Aturan cramer"},"content":{"rendered":"<p>Di halaman ini Anda akan melihat apa itu aturan Cramer dan, sebagai tambahan, Anda akan menemukan contoh dan latihan penyelesaian sistem persamaan menggunakan aturan Cramer.<\/p>\n<h2 class=\"wp-block-heading\"> Apa aturan Cramer?<\/h2>\n<p> <strong>Aturan Cramer<\/strong> adalah metode yang digunakan untuk menyelesaikan sistem persamaan dengan determinan. Mari kita lihat cara penggunaannya:<\/p>\n<p> Pertimbangkan sistem persamaan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e0141f3451719f665ef28e4061489551_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} ax+by+cz= \\color{red}\\bm{j} \\\\[1.5ex] dx+ey+fz=\\color{red}\\bm{k} \\\\[1.5ex] gx+hy+iz = \\color{red}\\bm{l} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"171\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Matriks A dan matriks perluasan A&#8217; dari sistem tersebut adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1d628a13ec7de4b3ba7a301c0a5d8ac6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} a &amp; b &amp; c  \\\\[1.1ex] d &amp; e &amp; f  \\\\[1.1ex] g &amp; h &amp; i  \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} a &amp; b &amp; c &amp;  \\color{red}\\bm{j}  \\\\[1.1ex] d &amp; e &amp; f &amp; \\color{red}\\bm{k} \\\\[1.1ex] g &amp; h &amp; i &amp; \\color{red}\\bm{l} \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"384\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <strong><span style=\"text-decoration: underline;\">Aturan Cramer<\/span><\/strong> mengatakan bahwa solusi suatu sistem persamaan adalah: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/quelle-est-la-regle-de-cramer.webp\" alt=\"apa itu aturan cramer, penjelasan aturan cramer\" class=\"wp-image-1062\" width=\"677\" height=\"385\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Perhatikan bahwa determinan pembilangnya seperti determinan matriks A tetapi mengubah kolom masing-masing suku yang tidak diketahui menjadi kolom suku-suku bebas.<\/p>\n<p> Oleh karena itu, aturan Cramer digunakan untuk menyelesaikan sistem persamaan linier. Namun seperti yang telah anda ketahui, ada banyak cara untuk menyelesaikan suatu sistem persamaan, misalnya<a href=\"https:\/\/mathority.org\/id\/metode-jordan-gauss-dengan-contoh-dan-latihan-yang-diselesaikan\/\">metode Gauss Jordan<\/a> yang terkenal.<\/p>\n<p> Di bawah ini adalah contoh penyelesaian sistem persamaan linear dengan aturan Cramer, atau terkadang juga ditulis dengan aturan Kramer.<\/p>\n<h2 class=\"wp-block-heading\"> Contoh 1: sistem kompatibel yang ditentukan (SCD)<\/h2>\n<ul>\n<li> Selesaikan sistem 3 persamaan berikut dengan 3 variabel yang tidak diketahui menggunakan aturan Cramer:<\/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-6013b7e73c89c24fe388f1a5d018f32b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 2x+y+3z= 1 \\\\[1.5ex] 3x-2y-z=0 \\\\[1.5ex] x+3y+2z = 5\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"135\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Pertama-tama kita buat matriks A dan matriks perluasan A&#8217; dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c710ed86223f47f39b5a25720b5ca19d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 2 &amp; 1 &amp; 3 \\\\[1.1ex] 3 &amp; -2 &amp; -1 \\\\[1.1ex] 1 &amp; 3 &amp; 2\\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 2 &amp; 1 &amp; 3 &amp; 1 \\\\[1.1ex] 3 &amp; -2 &amp; -1 &amp; 0 \\\\[1.1ex] 1 &amp; 3 &amp; 2 &amp; 5 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"405\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita menghitung rank kedua matriks tersebut, untuk melihat jenis sistemnya. Untuk menghitung rank A, kita menghitung determinan 3\u00d73 dari seluruh matriks (menggunakan aturan Sarrus) dan melihat apakah hasilnya 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-ae4a3bb88d113494463df8e670c326c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 2 &amp; 1 &amp; 3 \\\\[1.1ex] 3 &amp; -2 &amp; -1 \\\\[1.1ex] 1 &amp; 3 &amp; 2\\end{vmatrix} =-8-1+27+6+6-6 = 24 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"427\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Penentu A berbeda dengan 0, sehingga <strong>matriks A mempunyai 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-842ae3b68df41813d9e409968f3ae946_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> Jadi <strong>matriks A&#8217; juga mempunyai rangking 3<\/strong> , karena matriks tersebut tidak boleh mempunyai rangking 4 dan paling sedikit harus mempunyai rangking yang sama dengan matriks A.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Luas matriks A sama dengan luas matriks A&#8217; dan jumlah sistem yang tidak diketahui (3), oleh karena itu, berdasarkan <strong>teorema Rouch\u00e9-Frobenius<\/strong> , kita mengetahui bahwa ini adalah <strong>sistem kompatibel yang ditentukan<\/strong> (SCD):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-557185e16670c72d23eec5a3ea13b487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 3 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = n = 3  \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCD}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"436\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Setelah kami mengetahui bahwa sistem tersebut adalah SCD, kami menerapkan <strong>aturan Cramer<\/strong> untuk menyelesaikannya. Untuk melakukannya, ingatlah bahwa matriks A, determinannya, dan matriks A&#8217; adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31b2b3e5865c2264c360fb887d37a5f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 2 &amp; 1 &amp; 3 \\\\[1.1ex] 3 &amp; -2 &amp; -1 \\\\[1.1ex] 1 &amp; 3 &amp; 2\\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 2 &amp; 1 &amp; 3 &amp; \\color{red}\\bm{1} \\\\[1.1ex] 3 &amp; -2 &amp; -1 &amp; \\color{red}\\bm{0} \\\\[1.1ex] 1 &amp; 3 &amp; 2 &amp; \\color{red}\\bm{5} \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"431\" 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-0a604d8f5a3927a47a264d28f7a007b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 2 &amp; 1 &amp; 3 \\\\[1.1ex] 3 &amp; -2 &amp; -1 \\\\[1.1ex] 1 &amp; 3 &amp; 2\\end{vmatrix} =24\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"187\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-757d0eed520b26d08cc3b8b397d0f980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita ubah kolom pertama determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a1fa494ffb5e452d59c4d2dad40f925a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x} = \\cfrac{\\begin{vmatrix} \\color{red}\\bm{1} &amp; 1 &amp; 3 \\\\[1.1ex] \\color{red}\\bm{0} &amp; -2 &amp; -1 \\\\[1.1ex] \\color{red}\\bm{5} &amp; 3 &amp; 2 \\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{24}{24} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"238\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5fb4fb8b1addff607711094fd1ed326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom kedua determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-08e3dabe2f33434eb96658491f67c0b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{y} = \\cfrac{\\begin{vmatrix} 2 &amp; \\color{red}\\bm{1} &amp; 3 \\\\[1.1ex] 3 &amp;  \\color{red}\\bm{0} &amp; -1 \\\\[1.1ex] 1 &amp; \\color{red}\\bm{5} &amp; 2\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{48}{24} = \\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"223\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Menghitung<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23aa090e6102a41de5ad5515112e4d03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  z\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom ketiga determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96e76cb8867224755e9c19254678abd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{z} = \\cfrac{\\begin{vmatrix} 2 &amp; 1 &amp; \\color{red}\\bm{1} \\\\[1.1ex] 3 &amp; -2 &amp;  \\color{red}\\bm{0} \\\\[1.1ex] 1 &amp; 3 &amp;  \\color{red}\\bm{5}\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{-24}{24} = \\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"259\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, penyelesaian sistem persamaan tersebut adalah:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-be5a19fed42dcb59880c2d0eee8e51f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x = 1 \\qquad y=2 \\qquad z = -1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"210\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Contoh 2: Sistem Kompatibel Tak tentu (ICS)<\/h2>\n<ul>\n<li> Selesaikan sistem persamaan berikut menggunakan aturan Cramer:<\/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-781530aac4d8507fd6c7cbd77c3b4651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 3x+2y+4z=1 \\\\[1.5ex] -2x+3y-z=0 \\\\[1.5ex] x+5y+3z = 1 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"149\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Pertama-tama kita buat matriks A dan matriks perluasan A&#8217; dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a64800a78bf8e2e2f547be907e6863cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 3 &amp; 2 &amp; 4 \\\\[1.1ex] -2 &amp; 3 &amp; -1 \\\\[1.1ex] 1 &amp; 5 &amp; 3 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 3 &amp; 2 &amp; 4 &amp; 1 \\\\[1.1ex] -2 &amp; 3 &amp; -1 &amp; 0 \\\\[1.1ex] 1 &amp; 5 &amp; 3 &amp; 1 \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"405\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita menghitung rentang kedua matriks dan dengan demikian dapat melihat jenis sistemnya. Untuk menghitung rank A, kita menghitung determinan seluruh matriks (menggunakan aturan Sarrus) dan memeriksa apakah hasilnya 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-581c58cbe0fdd9952e7e25b919ecc33b_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; 3 &amp; -1 \\\\[1.1ex] 1 &amp; 5 &amp; 3\\end{vmatrix} = 27-2-40-12+15+12= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"407\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Penentunya menghasilkan 0, sehingga matriks A tidak berpangkat 3. Tetapi matriks A mempunyai determinan 2\u00d72 yang berbeda dengan 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-a5d1acad8bc31240f80d8cfbf3605997_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; 2 \\\\[1.1ex] -2 &amp; 3 \\end{vmatrix} =9-(-4)=13\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"222\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Jadi <strong>matriks A 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Setelah kita mengetahui luas matriks A, kita menghitung luas matriks A&#8217;. Penentu dari 3 kolom pertama menghasilkan 0, jadi kita coba kemungkinan determinan 3\u00d73 lainnya pada matriks A&#8217;:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-686e7ca635ecee685005f6013c2e64ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 2 &amp; 4 &amp; 1 \\\\[1.1ex] 3 &amp; -1 &amp; 0 \\\\[1.1ex] 5 &amp; 3 &amp; 1 \\end{vmatrix} = 0 \\qquad \\begin{vmatrix} 3 &amp; 4 &amp; 1 \\\\[1.1ex] -2 &amp; -1 &amp; 0 \\\\[1.1ex] 1 &amp; 3 &amp; 1 \\end{vmatrix} = 0 \\qquad \\begin{vmatrix} 3 &amp; 2 &amp; 1 \\\\[1.1ex] -2 &amp; 3 &amp; 0 \\\\[1.1ex] 1 &amp; 5 &amp; 1 \\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"440\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Semua determinan berorde 3 menghasilkan 0. Namun, jelas bahwa matriks A&#8217; mempunyai determinan non-0 2\u00d72 yang sama dengan matriks A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a5d1acad8bc31240f80d8cfbf3605997_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; 2 \\\\[1.1ex] -2 &amp; 3 \\end{vmatrix} =9-(-4)=13\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"222\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, <strong>matriks A&#8217; juga 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-80398cfd2fff647f81c0d4160f3b2f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Jadi, karena pangkat matriks A sama dengan pangkat matriks A&#8217; tetapi keduanya lebih kecil dari jumlah sistem yang tidak diketahui (3), kita mengetahui melalui <strong>teorema Rouch\u00e9-Frobenius<\/strong> bahwa ini adalah <strong>Sistem yang Kompatibel Tak Pasti<\/strong> (ICS):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96868a2569ea0ab5ca99d8dc606d3dc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=2 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = 2 \\ < \\ n =3  \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"475\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-background\" style=\"background-color:#dff6ff\"> Ketika kita ingin menyelesaikan sistem tak tentu yang kompatibel (SCI), kita perlu <strong>mentransformasikan sistemnya<\/strong> : pertama-tama kita hilangkan persamaannya, kemudian kita ubah sebuah variabel menjadi \u03bb (biasanya variabel z), dan terakhir kita satukan suku-suku yang memiliki \u03bb dengan ketentuan independen.<\/p>\n<p> Setelah kita mentransformasikan sistem, kita menerapkan aturan Cramer dan kita akan mendapatkan solusi sistem sebagai fungsi dari \u03bb.<\/p>\n<p> Dalam hal ini, <strong>kita akan menghilangkan persamaan terakhir<\/strong> dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f0511fecc9c2af695b6b8eccae6b0661_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 3x+2y+4z=1 \\\\[1.5ex] -2x+3y-z=0 \\\\[1.5ex]\\cancel{x+5y+3z = 1} \\end{cases} \\longrightarrow \\quad \\begin{cases} 3x+2y+4z=1 \\\\[1.5ex] -2x+3y-z=0\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"377\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> <strong>Sekarang mari kita ubah variabel z menjadi \u03bb:<\/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-2d6142d2be611954fd849a032a97245a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 3x+2y+4z=1 \\\\[1.5ex] -2x+3y-z=0  \\end{cases} \\xrightarrow{z \\ = \\ \\lambda}\\quad \\begin{cases} 3x+2y+4\\lambda=1 \\\\[1.5ex] -2x+3y-\\lambda=0\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"398\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dan kami menempatkan <strong>suku dengan \u03bb dengan suku independen:<\/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-00214205f2334f1c9bc10810c1c1df83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 3x+2y=1-4\\lambda \\\\[1.5ex] -2x+3y=\\lambda \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Oleh karena itu, matriks A dan matriks A&#8217; dari sistem tersebut tetap:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c4b47303973b823a1c5628f5448ca79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 3 &amp; 2  \\\\[1.1ex] -2 &amp; 3 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{cc|c} 3 &amp; 2 &amp; 1 -4\\lambda \\\\[1.1ex] -2 &amp; 3 &amp; \\lambda \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"363\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Terakhir, setelah kita mengubah sistem, <strong>kita menerapkan aturan Cramer<\/strong> . Oleh karena itu, kita menyelesaikan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d1b79f52dc82f5cfc311867273e78c06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 3 &amp; 2  \\\\[1.1ex] -2 &amp; 3\\end{vmatrix} = 13\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"148\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-757d0eed520b26d08cc3b8b397d0f980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita ubah kolom pertama determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0ff917eaea976c65bd18e0476078d3cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x} = \\cfrac{\\begin{vmatrix} 1 -4\\lambda &amp; 2  \\\\[1.1ex] \\lambda &amp; 3 \\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{3(1-4\\lambda) -2\\lambda}{13} = \\cfrac{\\bm{3-14\\lambda} }{\\bm{13}}\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"349\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5fb4fb8b1addff607711094fd1ed326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom kedua determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-155ca520739bbf7e040a6cdc632f7c27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{y} = \\cfrac{\\begin{vmatrix} 3 &amp; 1 -4\\lambda  \\\\[1.1ex]-2&amp;  \\lambda  \\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{3\\lambda -\\bigl(-2(1-4\\lambda)\\bigr)}{13}= \\cfrac{3\\lambda -\\bigl(-2+8\\lambda\\bigr)}{13} = \\cfrac{\\bm{2-5\\lambda} }{\\bm{13}}\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"529\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Meskipun penyelesaian sistem persamaan tersebut adalah fungsi dari \u03bb, karena merupakan SCI dan oleh karena itu, ia mempunyai banyak penyelesaian yang tak terhingga:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c9866e045041eb2d8fe103db2309f229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x =} \\cfrac{\\bm{3-14\\lambda} }{\\bm{13}} \\qquad \\bm{y=}\\cfrac{\\bm{2-5\\lambda} }{\\bm{13}} \\qquad \\bm{z = \\lambda}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"283\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Aturan Cramer Memecahkan Masalah <\/h2>\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 1<\/h3>\n<p> Terapkan aturan Cramer untuk menyelesaikan sistem dua persamaan berikut dengan 2 persamaan yang tidak diketahui: <\/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\/exercice-resolu-pas-a-pas-de-la-regle-de-cramer-22.webp\" alt=\"latihan diselesaikan langkah demi langkah dengan aturan 2x2 Cramer\" class=\"wp-image-3999\" width=\"137\" height=\"83\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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\"> Hal pertama yang harus dilakukan adalah matriks A dan matriks perluasan A&#8217; dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a001db9cf56846150730fee7126dacd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{cc} 2 &amp; 5 \\\\[1.1ex] 1 &amp; 4 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{cc|c} 2 &amp; 5 &amp; 8 \\\\[1.1ex] 1 &amp; 4 &amp; 7 \\end{array}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"294\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita harus mencari rank matriks A. Untuk melakukannya, kita periksa apakah determinan seluruh matriks berbeda dari 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-0c75c1c344c286016bea83237f1f418e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 2 &amp; 5 \\\\[1.1ex] 1 &amp; 4 \\end{vmatrix} = 8-5=3 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena matriks mempunyai determinan 2\u00d72 yang berbeda dengan 0, <strong>maka matriks A mempunyai rank 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah kita mengetahui pangkat A, kita menghitung pangkat A&#8217;. Ini paling tidak berada pada peringkat 2, karena kita baru saja melihat bahwa di dalamnya terdapat determinan berorde 2 yang berbeda dari 0. Selain itu, ia tidak dapat berada pada peringkat 3, karena kita tidak dapat tidak membuat determinan 3\u00d73. Oleh karena itu, <strong>matriks A&#8217; juga 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-80398cfd2fff647f81c0d4160f3b2f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, dengan menerapkan <strong>teorema Rouch\u00e9-Frobenius,<\/strong> kita mengetahui bahwa ini adalah <strong>sistem determinasi yang kompatibel<\/strong> (SCD), karena jangkauan A sama dengan jangkauan A&#8217; dan jumlah yang tidak diketahui.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bbd67b16bb6d52a0696e70a77833cd3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=2 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 2 \\end{array}} \\\\ \\\\ \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = n = 2 \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCD}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"436\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah kami mengetahui bahwa sistem tersebut adalah SCD, kami menerapkan <strong>aturan Cramer<\/strong> untuk menyelesaikannya.<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-757d0eed520b26d08cc3b8b397d0f980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita ubah kolom pertama determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b0adeda8f2ce557661466996038b1148_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x} = \\cfrac{\\begin{vmatrix} 8 &amp; 5 \\\\[1.1ex] 7 &amp; 4\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{-3}{3} = \\bm{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"186\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5fb4fb8b1addff607711094fd1ed326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom kedua determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-59790a66cc31fac07be1d5a7bb556d9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{y} = \\cfrac{\\begin{vmatrix}2 &amp; 8 \\\\[1.1ex] 1 &amp; 7\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{6}{3} = \\bm{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"150\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, penyelesaian sistem persamaan tersebut adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26fa7c9ed2d05ca07ff62a968ba7ab11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x = -1 \\qquad y=2}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"133\" style=\"vertical-align: -4px;\"><\/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> Temukan solusi dari sistem tiga persamaan berikut dengan 3 yang tidak diketahui menggunakan aturan Cramer: <\/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\/exercice-resolu-du-systeme-de-regles-de-cramer-des-equations-3-3.webp\" alt=\"Latihan terpecahkan aturan Cramer tentang sistem persamaan 3x3\" class=\"wp-image-4002\" width=\"181\" height=\"124\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 buat matriks A dan matriks perluasan A&#8217; dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-eea75fbf6d86ebc3d0b9e236cd2160f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 1 &amp; 3 &amp; 2\\\\[1.1ex] -1 &amp; 5 &amp; -1\\\\[1.1ex] 3 &amp; -1 &amp; 4 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 1 &amp; 3 &amp; 2 &amp; 2 \\\\[1.1ex] -1 &amp; 5 &amp; -1 &amp; 4 \\\\[1.1ex] 3 &amp; -1 &amp; 4 &amp; 0 \\end{array}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"432\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita mencari rank matriks A dengan menghitung determinan matriks 3\u00d73 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-73f751f3b5c527c16b5de1b10bf07a4e_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] -1 &amp; 5 &amp; -1\\\\[1.1ex] 3 &amp; -1 &amp; 4 \\end{vmatrix} = 20-9+2-30-1+12=-6 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"445\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Matriks yang mempunyai determinan orde 3 berbeda dengan 0, <strong>matriks A mempunyai rank 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-842ae3b68df41813d9e409968f3ae946_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 class=\"has-text-align-left\"> akibatnya, matriks A&#8217; juga menduduki peringkat 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-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, dengan menggunakan <strong>teorema Rouch\u00e9-Frobenius,<\/strong> kita mengetahui bahwa ini adalah <strong>sistem determinasi yang kompatibel<\/strong> (SCD), karena jangkauan A sama dengan jangkauan A&#8217; dan jumlah yang tidak diketahui.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31b495a48a75d7af1f23e38818bf4eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 3 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3 \\end{array}} \\\\ \\\\ \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = n = 3 \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCD}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"436\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah kita mengetahui bahwa sistem tersebut adalah SCD, kita perlu menerapkan <strong>aturan Cramer<\/strong> untuk menyelesaikan sistem tersebut.<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-757d0eed520b26d08cc3b8b397d0f980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita ubah kolom pertama determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fc574297f609b68e4fb48466ec6c8077_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x} = \\cfrac{\\begin{vmatrix} 2 &amp; 3 &amp; 2 \\\\[1.1ex] 4 &amp; 5 &amp; -1\\\\[1.1ex]0 &amp; -1 &amp; 4\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{-18}{-6} = \\bm{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"235\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5fb4fb8b1addff607711094fd1ed326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom kedua determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2544601137d62e217ff1866f278203d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{y} = \\cfrac{\\begin{vmatrix}1 &amp; 2 &amp; 2 \\\\[1.1ex] -1 &amp; 4 &amp; -1\\\\[1.1ex] 3 &amp; 0 &amp; 4\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{-6}{-6} = \\bm{1}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"224\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Menghitung<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23aa090e6102a41de5ad5515112e4d03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  z\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom ketiga determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42d7d4adcfc48954185ca14b56b8e128_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{z} = \\cfrac{\\begin{vmatrix} 1 &amp; 3 &amp; 2 \\\\[1.1ex] -1 &amp; 5 &amp; 4 \\\\[1.1ex] 3 &amp; -1 &amp; 0\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{12}{-6} = \\bm{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"230\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, penyelesaian sistem persamaan tersebut adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-685195d3a299f30f6421bb387f7f00e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x =3 \\qquad y=1 \\qquad z=-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"210\" style=\"vertical-align: -4px;\"><\/p>\n<\/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 solusi sistem tiga persamaan berikut dengan 3 yang tidak diketahui menggunakan aturan Cramer: <\/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\/exemple-de-regle-de-cramer.webp\" alt=\"contoh aturan Cramer\" class=\"wp-image-4003\" width=\"183\" height=\"123\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 buat matriks A dan matriks perluasan A&#8217; dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-afd359275e5ebaaf3229504c47a5815f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 1 &amp; 2 &amp; 5\\\\[1.1ex] 2 &amp; 3 &amp; -1 \\\\[1.1ex] 3 &amp; 4 &amp; -7 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 1 &amp; 2 &amp; 5 &amp; 1 \\\\[1.1ex] 2 &amp; 3 &amp; -1 &amp; 5 \\\\[1.1ex] 3 &amp; 4 &amp; -7 &amp; 9 \\end{array}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"377\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung luas matriks A: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-47ddf17a2b3eed5a680d685900a79b31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 1 &amp; 2 &amp; 5\\\\[1.1ex] 2 &amp; 3 &amp; -1 \\\\[1.1ex] 3 &amp; 4 &amp; -7 \\end{vmatrix} =-21-6+40-45+4+28=0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"398\" 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-fdd4380c7c76418bd3ec12c94359f886_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 2 \\\\[1.1ex] 2 &amp; 3  \\end{vmatrix} = 3-4 = -1 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"186\" 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah kita mengetahui luas matriks A, kita menghitung luas matriks A&#8217;. Penentu dari 3 kolom pertama menghasilkan 0, jadi kita coba kemungkinan determinan 3\u00d73 lainnya pada matriks A&#8217;:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1addc62130e0462075b3bade26a7e35e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 2 &amp; 5 &amp; 1 \\\\[1.1ex]  3 &amp; -1 &amp; 5 \\\\[1.1ex] 4 &amp; -7 &amp; 9 \\end{vmatrix} = 0 \\qquad \\begin{vmatrix} 1 &amp; 5 &amp; 1 \\\\[1.1ex] 2 &amp; -1 &amp; 5 \\\\[1.1ex] 3 &amp; -7 &amp; 9\\end{vmatrix} = 0 \\qquad \\begin{vmatrix} 1 &amp; 2 &amp; 1 \\\\[1.1ex] 2 &amp; 3 &amp; 5 \\\\[1.1ex] 3 &amp; 4 &amp; 9 \\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"412\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Semua determinan berorde 3 menghasilkan 0. Namun matriks A&#8217; mempunyai determinan non-0 2\u00d72 yang sama dengan matriks A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7de377466bd5afd03f58f9b532324e75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 1 &amp; 2 \\\\[1.1ex] 2 &amp; 3 \\end{vmatrix} = 3-4 = -1 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"186\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, matriks A&#8217; juga mempunyai rangking 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-80398cfd2fff647f81c0d4160f3b2f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Karena pangkat matriks A sama dengan pangkat matriks A&#8217; tetapi keduanya lebih kecil dari jumlah sistem yang tidak diketahui (3), kita mengetahui melalui <strong>teorema Rouch\u00e9-Frobenius<\/strong> bahwa ini adalah <strong>Sistem Kompatibel Tak tentu<\/strong> (ICS):<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96868a2569ea0ab5ca99d8dc606d3dc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=2 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = 2 \\ < \\ n =3  \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"475\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sebagai sistem ICS, kita harus menghilangkan persamaan. Dalam hal ini, <strong>kita akan menghilangkan persamaan terakhir<\/strong> dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a1d067e155540f4345cf56e5c1567d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} x+2y+5z=1 \\\\[1.5ex] 2x+3y-z=5 \\\\[1.5ex]\\cancel{3x+4y-7z = 9} \\end{cases} \\longrightarrow \\quad \\begin{cases} x+2y+5z=1 \\\\[1.5ex] 2x+3y-z=5\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"357\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Sekarang mari kita ubah variabel z menjadi \u03bb:<\/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-b5fa91777a722d3783b2f887aab44152_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} x+2y+5z=1 \\\\[1.5ex] 2x+3y-z=5  \\end{cases} \\xrightarrow{z \\ = \\ \\lambda}\\quad \\begin{cases} x+2y+5\\lambda=1 \\\\[1.5ex] 2x+3y-\\lambda=5\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"369\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kami menempatkan suku dengan \u03bb dengan suku independen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76ff21181be050b01c247981298986a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} x+2y=1-5\\lambda\\\\[1.5ex] 2x+3y=5+\\lambda \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sehingga matriks A dan matriks A&#8217; sistem tetap:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-230e5b28dd467127e63f4f9756cf90da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 1 &amp; 2  \\\\[1.1ex] 2 &amp; 3 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{cc|c} 1 &amp; 2 &amp; 1 -5\\lambda \\\\[1.1ex] 2 &amp; 3 &amp;5+\\lambda \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"335\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, setelah kita mengubah sistem, <strong>kita menerapkan aturan Cramer<\/strong> . Oleh karena itu, kita menyelesaikan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f127efbd217e2bca8852ec792610732f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 1 &amp; 2 \\\\[1.1ex] 2 &amp; 3\\end{vmatrix} =-1\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"138\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-757d0eed520b26d08cc3b8b397d0f980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita ubah kolom pertama determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42652a14362b42e606841b6bb3e77cc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x} = \\cfrac{\\begin{vmatrix} 1-5\\lambda &amp; 2 \\\\[1.1ex] 5+\\lambda &amp; 3 \\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{3-15\\lambda -(10+2\\lambda)}{-1} = \\cfrac{-7-17\\lambda}{-1} = \\bm{7+17\\lambda}\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"491\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5fb4fb8b1addff607711094fd1ed326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom kedua determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b95c5870f1762a2d82c9ebcccbca7408_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{y} = \\cfrac{\\begin{vmatrix} 1 &amp; 1-5\\lambda \\\\[1.1ex] 2 &amp; 5+\\lambda \\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{5+\\lambda -(2-10\\lambda)}{-1}= \\cfrac{3+11\\lambda}{-1} = \\bm{-3-11\\lambda}\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"465\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Meskipun penyelesaian sistem persamaan tersebut adalah fungsi dari \u03bb, karena merupakan SCI dan oleh karena itu, ia mempunyai banyak penyelesaian yang tak terhingga: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5483357f081aca551b07fe7c8f9ebf5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x =7+17\\lambda} \\qquad \\bm{y=-3-11\\lambda} \\qquad \\bm{z = \\lambda}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"311\" style=\"vertical-align: -4px;\"><\/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-119\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Latihan 4<\/h3>\n<p> Selesaikan soal sistem tiga persamaan dengan 3 variabel tak diketahui berikut ini dengan menerapkan aturan Cramer: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-61e1c3458f33b863db10750b9e51d09e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} -2x+5y+z=8 \\\\[1.5ex] 6x+2y+4z=4 \\\\[1.5ex] 3x-2y+z = -2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"149\" 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 membuat matriks A dan matriks perluasan A&#8217; dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-743a40010cb4a610e8a3fc6ae5d313b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc}-2 &amp; 5 &amp; 1 \\\\[1.1ex] 6 &amp; 2 &amp; 4 \\\\[1.1ex] 3 &amp; -2 &amp; 1\\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} -2 &amp; 5 &amp; 1 &amp; 8 \\\\[1.1ex] 6 &amp; 2 &amp; 4 &amp; 4 \\\\[1.1ex] 3 &amp; -2 &amp; 1 &amp; -2 \\end{array}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"419\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita hitung rank matriks A dengan menghitung determinan matriks 3&#215;3 menggunakan 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-713c634fbc3e1b1cb228e3891c9bff1c_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] 6 &amp; 2 &amp; 4 \\\\[1.1ex] 3 &amp; -2 &amp; 1 \\end{vmatrix} = -4+60-12-6-16-30=-8 \\bm{\\neq 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"453\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Matriks yang mempunyai determinan orde 3 berbeda dengan 0, <strong>matriks A mempunyai rank 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-842ae3b68df41813d9e409968f3ae946_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 class=\"has-text-align-left\"> akibatnya matriks A&#8217; juga mempunyai rangking 3, karena paling sedikit harus mempunyai rangking yang sama dengan matriks A dan tidak boleh mempunyai rangking 4 karena matriks tersebut berdimensi 3\u00d74.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-150bbc9c8e363db471c2d5bc4f33e1fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, dengan menggunakan <strong>teorema Rouch\u00e9-Frobenius,<\/strong> kami menyimpulkan bahwa ini adalah <strong>sistem yang kompatibel dengan determinasi<\/strong> (SCD), karena jangkauan A sama dengan jangkauan A&#8217; dan jumlah yang tidak diketahui.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-31b495a48a75d7af1f23e38818bf4eca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 3 \\\\[1.3ex] \\color{black}rg(A')=3 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3 \\end{array}} \\\\ \\\\ \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = n = 3 \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCD}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"436\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah kita mengetahui bahwa sistem tersebut adalah SCD, kita perlu menerapkan <strong>aturan Cramer<\/strong> untuk menyelesaikan sistem tersebut.<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-757d0eed520b26d08cc3b8b397d0f980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita ubah kolom pertama determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8a290479c69ff806f19dcf29f96e1228_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x} = \\cfrac{\\begin{vmatrix} 8 &amp; 5 &amp; 1 \\\\[1.1ex] 4 &amp; 2 &amp; 4 \\\\[1.1ex] -2 &amp; -2 &amp; 1\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{16}{-8} = \\bm{-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"231\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5fb4fb8b1addff607711094fd1ed326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom kedua determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8bba0765fbcbcebf0585520af25b4a30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{y} = \\cfrac{\\begin{vmatrix}-2 &amp; 8 &amp; 1 \\\\[1.1ex] 6 &amp; 4 &amp; 4 \\\\[1.1ex] 3 &amp; -2 &amp; 1\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{0}{-6} = \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"217\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Menghitung<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-23aa090e6102a41de5ad5515112e4d03_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  z\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom ketiga determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5bc157a8c4dfe8ee4651affac68ef878_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{z} = \\cfrac{\\begin{vmatrix} -2 &amp; 5 &amp; 8 \\\\[1.1ex] 6 &amp; 2 &amp; 4 \\\\[1.1ex] 3 &amp; -2 &amp; -2\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{-32}{-8} = \\bm{4}\" title=\"Rendered by QuickLaTeX.com\" height=\"113\" width=\"247\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, penyelesaian sistem persamaan linear adalah: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c004c5c466235d2d1a784707145d952_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x =-2 \\qquad y=0 \\qquad z=4}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"211\" style=\"vertical-align: -4px;\"><\/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> Selesaikan sistem persamaan linear berikut menggunakan aturan Cramer: <\/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\/exemple-comment-resoudre-un-systeme-dequations-avec-la-regle-de-cramer.webp\" alt=\"Contoh penyelesaian sistem persamaan dengan aturan Cramer\" class=\"wp-image-4008\" width=\"215\" height=\"127\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 buat matriks A dan matriks perluasan A&#8217; dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f5153b5951b768cc3cafa2bb2567ba92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 3 &amp; -2 &amp; -3 \\\\[1.1ex] -1 &amp; 5 &amp; 4 \\\\[1.1ex] 5 &amp; 1 &amp; -2 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{ccc|c} 3 &amp; -2 &amp; -3 &amp; 4 \\\\[1.1ex] -1 &amp; 5 &amp; 4 &amp; -10 \\\\[1.1ex] 5 &amp; 1 &amp; -2 &amp; -2 \\end{array}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"455\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung luas matriks A: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3778c9499e2a44ea3834dfed1523163_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix} 3 &amp; -2 &amp; -3 \\\\[1.1ex] -1 &amp; 5 &amp; 4 \\\\[1.1ex] 5 &amp; 1 &amp; -2 \\end{vmatrix} =-30-40+3+75-12+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"426\" 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-03d70742b14ced92f33963df0c86e92f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; -2 \\\\[1.1ex] -1 &amp; 5  \\end{vmatrix} = 15- (2)= 13 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"231\" 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-eded270b78ab3d95ce827e3ea428efb1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A)=2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Setelah kita mengetahui luas matriks A, kita menghitung luas matriks A&#8217;. Penentu dari 3 kolom pertama menghasilkan 0, jadi kita coba kemungkinan determinan 3\u00d73 lainnya pada matriks A&#8217;:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5bed93d532ae4ccd4649a73662f55f0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} -2 &amp; -3 &amp; 4 \\\\[1.1ex] 5 &amp; 4 &amp; -10 \\\\[1.1ex]  1 &amp; -2 &amp; -2 \\end{vmatrix} = 0 \\qquad \\begin{vmatrix}3 &amp; -3 &amp; 4 \\\\[1.1ex] -1 &amp; 4 &amp; -10 \\\\[1.1ex] 5 &amp; -2 &amp; -2\\end{vmatrix} = 0 \\qquad \\begin{vmatrix} 3 &amp; -2 &amp; 4 \\\\[1.1ex] -1 &amp; 5 &amp; -10 \\\\[1.1ex] 5 &amp; 1 &amp;-2\\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"535\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Semua determinan berorde 3 menghasilkan 0. Namun, jelas matriks A&#8217; mempunyai determinan berorde 2 selain 0 yang sama dengan matriks A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-858d95d7d252b16706b66c0e6aba09c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix} 3 &amp; -2 \\\\[1.1ex] -1 &amp; 5 \\end{vmatrix} = 13 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, matriks A&#8217; juga mempunyai rangking 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-80398cfd2fff647f81c0d4160f3b2f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  rg(A')=2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"81\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Pangkat matriks A sama dengan pangkat matriks A&#8217; tetapi keduanya lebih kecil dari jumlah yang tidak diketahui dari sistem (3), sehingga berdasarkan <strong>teorema Rouch\u00e9-Frobenius<\/strong> kita mengetahui bahwa matriks tersebut merupakan <strong>Indeterminate System Kompatibel<\/strong> (SCI) :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-96868a2569ea0ab5ca99d8dc606d3dc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{array}{c} \\begin{array}{c} \\color{black}rg(A) = 2 \\\\[1.3ex] \\color{black}rg(A')=2 \\\\[1.3ex] \\color{black}\\text{N\\'umero de inc\\'ognitas} = 3    \\end{array}} \\\\ \\\\  \\color{blue} \\boxed{ \\color{black}\\phantom{^9_9} rg(A) = rg(A') = 2 \\ < \\ n =3  \\color{blue} \\ \\bm{\\longrightarrow} \\ \\color{black} \\bm{SCI}\\phantom{^9_9}} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"138\" width=\"475\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sebagai sistem ICS, kita harus menghilangkan satu persamaan. Dalam hal ini, <strong>kita akan menghilangkan persamaan terakhir<\/strong> dari sistem:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e10bd826663dff41c4272610cbc07b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 3x-2y-3z=4 \\\\[1.5ex] -x+5y+4z=-10 \\\\[1.5ex]\\cancel{5x+y-2z = -2} \\end{cases} \\longrightarrow \\quad \\begin{cases} 3x-2y-3z=4 \\\\[1.5ex] -x+5y+4z=-10\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"423\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> <strong>Sekarang mari kita ubah variabel z menjadi \u03bb:<\/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-2502be450040b38761c08e5d6beaf379_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 3x-2y-3z=4 \\\\[1.5ex] -x+5y+4z=-10  \\end{cases} \\xrightarrow{z \\ = \\ \\lambda}\\quad \\begin{cases} 3x-2y-3\\lambda=4 \\\\[1.5ex] -x+5y+4\\lambda=-10\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"444\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kami menempatkan suku dengan \u03bb dengan suku independen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-80a43d98e6be30965d554e8a89aa5d89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{cases} 3x-2y=4+3\\lambda \\\\[1.5ex] -x+5y=-10-4\\lambda\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sehingga matriks A dan matriks A&#8217; sistem tetap:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3451ce571163983cf41794d4998283d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  A= \\left( \\begin{array}{ccc} 3 &amp; -2  \\\\[1.1ex] -1 &amp; 5 \\end{array} \\right) \\qquad A'= \\left( \\begin{array}{cc|c} 3 &amp; -2 &amp; 4+3\\lambda \\\\[1.1ex] 1 &amp; 5 &amp;-10-4\\lambda \\end{array} \\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"399\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Terakhir, setelah kita mengubah sistem, <strong>kita menerapkan aturan Cramer<\/strong> . Oleh karena itu, kita menyelesaikan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0e7a7d6208ea5e762f5c74a44e6838cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\begin{vmatrix}A \\end{vmatrix}= \\begin{vmatrix}3&amp; -2 \\\\[1.1ex] -1 &amp; 5\\end{vmatrix} =13\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"162\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-757d0eed520b26d08cc3b8b397d0f980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dengan aturan Cramer, kita ubah kolom pertama determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8c30fcc0526c2d4112eb4f60a3d8847f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x} = \\cfrac{\\begin{vmatrix} 4+3\\lambda &amp; -2 \\\\[1.1ex]-10-4\\lambda &amp; 5\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{20+15\\lambda -(20+8\\lambda)}{13} = \\cfrac{\\bm{7\\lambda}}{\\bm{13}}\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"394\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Untuk menghitung hal yang tidak diketahui<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5fb4fb8b1addff607711094fd1ed326e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> Dengan aturan Cramer, kita mengubah kolom kedua determinan A dengan kolom suku bebas dan membaginya dengan determinan A:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fdb22a54274e019c811c9051502c474a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{y} = \\cfrac{\\begin{vmatrix} 3 &amp; 4+3\\lambda \\\\[1.1ex] -1 &amp; -10-4\\lambda\\end{vmatrix}}{\\begin{vmatrix} A \\end{vmatrix}} = \\cfrac{-30-12\\lambda -(-4-3\\lambda)}{13}= \\cfrac{\\bm{-26-9\\lambda}}{\\bm{13}}\" title=\"Rendered by QuickLaTeX.com\" height=\"81\" width=\"473\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi, solusi sistem persamaan adalah fungsi dari \u03bb, karena merupakan SCI dan, oleh karena itu, sistem tersebut memiliki banyak solusi yang tak terhingga:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d0e525b9aca6bd683491ab7950f039e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  \\bm{x=} \\cfrac{\\bm{7\\lambda}}{\\bm{13}} \\qquad \\bm{y=} \\cfrac{\\bm{-26-9\\lambda}}{\\bm{13}} \\qquad \\bm{z = \\lambda}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"266\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Di halaman ini Anda akan melihat apa itu aturan Cramer dan, sebagai tambahan, Anda akan menemukan contoh dan latihan penyelesaian sistem persamaan menggunakan aturan Cramer. Apa aturan Cramer? Aturan Cramer adalah metode yang digunakan untuk menyelesaikan sistem persamaan dengan determinan. Mari kita lihat cara penggunaannya: Pertimbangkan sistem persamaan: Matriks A dan matriks perluasan A&#8217; dari &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/\"> <span class=\"screen-reader-text\">Aturan cramer<\/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":[51],"tags":[],"class_list":["post-305","post","type-post","status-publish","format-standard","hentry","category-sistem-pendidikan"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Aturan Cramer - 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-aturan-dan-latihan-cramer-yang-diselesaikan\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Aturan Cramer - Mathority\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan melihat apa itu aturan Cramer dan, sebagai tambahan, Anda akan menemukan contoh dan latihan penyelesaian sistem persamaan menggunakan aturan Cramer. Apa aturan Cramer? Aturan Cramer adalah metode yang digunakan untuk menyelesaikan sistem persamaan dengan determinan. Mari kita lihat cara penggunaannya: Pertimbangkan sistem persamaan: Matriks A dan matriks perluasan A&#8217; dari &hellip; Aturan cramer Selengkapnya &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T14:13:39+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e0141f3451719f665ef28e4061489551_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=\"9 menit\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Aturan cramer\",\"datePublished\":\"2023-07-06T14:13:39+00:00\",\"dateModified\":\"2023-07-06T14:13:39+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/\"},\"wordCount\":1771,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/id\/#organization\"},\"articleSection\":[\"Sistem pendidikan\"],\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/\",\"url\":\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/\",\"name\":\"Aturan Cramer - Mathority\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/#website\"},\"datePublished\":\"2023-07-06T14:13:39+00:00\",\"dateModified\":\"2023-07-06T14:13:39+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/#breadcrumb\"},\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/id\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Aturan cramer\"}]},{\"@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":"Aturan Cramer - 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-aturan-dan-latihan-cramer-yang-diselesaikan\/","og_locale":"id_ID","og_type":"article","og_title":"Aturan Cramer - Mathority","og_description":"Di halaman ini Anda akan melihat apa itu aturan Cramer dan, sebagai tambahan, Anda akan menemukan contoh dan latihan penyelesaian sistem persamaan menggunakan aturan Cramer. Apa aturan Cramer? Aturan Cramer adalah metode yang digunakan untuk menyelesaikan sistem persamaan dengan determinan. Mari kita lihat cara penggunaannya: Pertimbangkan sistem persamaan: Matriks A dan matriks perluasan A&#8217; dari &hellip; Aturan cramer Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/","article_published_time":"2023-07-06T14:13:39+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e0141f3451719f665ef28e4061489551_l3.png"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"9 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Aturan cramer","datePublished":"2023-07-06T14:13:39+00:00","dateModified":"2023-07-06T14:13:39+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/"},"wordCount":1771,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"articleSection":["Sistem pendidikan"],"inLanguage":"id","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/","url":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/","name":"Aturan Cramer - Mathority","isPartOf":{"@id":"https:\/\/mathority.org\/id\/#website"},"datePublished":"2023-07-06T14:13:39+00:00","dateModified":"2023-07-06T14:13:39+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/#breadcrumb"},"inLanguage":"id","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/id\/contoh-aturan-dan-latihan-cramer-yang-diselesaikan\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/id\/"},{"@type":"ListItem","position":2,"name":"Aturan cramer"}]},{"@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\/305","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=305"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts\/305\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/media?parent=305"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/categories?post=305"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/tags?post=305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}