{"id":331,"date":"2023-07-06T06:35:36","date_gmt":"2023-07-06T06:35:36","guid":{"rendered":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/"},"modified":"2023-07-06T06:35:36","modified_gmt":"2023-07-06T06:35:36","slug":"menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/","title":{"rendered":"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks"},"content":{"rendered":"<p>Pada halaman ini kami menjelaskan apa itu nilai eigen dan vektor eigen, disebut juga nilai eigen dan vektor eigen. Anda juga akan menemukan contoh cara menghitungnya serta latihan langkah demi langkah untuk dipraktikkan.<\/p>\n<h2 class=\"wp-block-heading\"> Apa yang dimaksud dengan nilai eigen dan vektor eigen?<\/h2>\n<p> Walaupun pengertian nilai eigen dan vektor eigen sulit dipahami, namun definisinya adalah sebagai berikut: <\/p>\n<div style=\"background-color:#dff6ff;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p style=\"text-align:left\"> <strong>Vektor eigen atau vektor eigen<\/strong> adalah vektor bukan nol pada peta linier yang bila ditransformasikan akan menimbulkan kelipatan skalar (tidak berubah arah). Skalar ini adalah <strong>nilai eigen atau nilai eigen<\/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-710a5e2df8739c35c060f790f5592734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Av = \\lambda v\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"65\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Emas<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah matriks peta linier,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef71511c70f0e4b25cc6bd69f3bc20c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"v\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah vektor eigen dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b5c45836864531b8e37025dabadd24a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> nilai sendiri.<\/p>\n<\/div>\n<p> Nilai eigen juga dikenal sebagai nilai karakteristik. Dan bahkan ada ahli matematika yang menggunakan akar kata Jerman \u201ceigen\u201d untuk menunjuk nilai eigen dan vektor eigen: <em>nilai eigen<\/em> untuk nilai eigen dan <em>vektor eigen<\/em> untuk vektor eigen.<\/p>\n<h2 class=\"wp-block-heading\"> Bagaimana cara menghitung nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) suatu matriks?<\/h2>\n<p> Untuk mencari nilai eigen dan vektor eigen suatu matriks, Anda harus mengikuti seluruh prosedur:<\/p>\n<ol style=\"color:#1976d2; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#262626;font-weight: normal;\">Persamaan karakteristik matriks dihitung dengan menyelesaikan determinan berikut:<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7224fcfc13d25429e22216a3d4124cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"92\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:15px\"> <span style=\"color:#262626;font-weight: normal;\">Kami menemukan akar polinomial karakteristik yang diperoleh pada langkah 1. Akar-akar ini adalah nilai eigen dari matriks.<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fbe85bd9aff702c72a31d3889f035518_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)=0 \\ \\longrightarrow \\ \\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"186\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<li style=\"margin-bottom:15px\"> <span style=\"color:#262626;font-weight: normal;\">Vektor eigen dari setiap nilai eigen dihitung. Untuk melakukan hal ini, sistem persamaan berikut diselesaikan untuk setiap nilai eigen:<\/span><\/li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-372f1009cb2b47f939cf9291f0f23885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-\\lambda I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/ol>\n<p> Ini adalah cara mencari nilai eigen dan vektor eigen suatu matriks, namun berikut kami juga memberikan beberapa tipsnya: \ud83d\ude09 <\/p>\n<div style=\"background-color:#fffde7;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px\" class=\"has-background\">\n<p style=\"text-align:left\"> <strong>Tips<\/strong> : kita dapat memanfaatkan sifat-sifat nilai eigen dan vektor eigen untuk menghitungnya dengan lebih mudah:<\/p>\n<p style=\"text-align:left\"> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong> Jejak matriks (jumlah diagonal utamanya) sama dengan jumlah seluruh nilai eigen.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4b8ae7f7f7a36be08403ae6ba8b3d32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle tr(A)=\\sum_{i=1}^n \\lambda_i\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"109\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong> Hasil kali semua nilai eigen sama dengan determinan matriks.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7aa4b68759894e3f25d6475c3b6f71b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle det(A)=\\prod_{i=1}^n \\lambda_i\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"115\" style=\"vertical-align: -21px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> <strong><span style=\"color:#1976d2;\">\u2713<\/span><\/strong> Jika terdapat kombinasi linier antar baris atau kolom, paling sedikit salah satu nilai eigen matriksnya sama dengan 0.<\/p>\n<\/div>\n<p> Mari kita lihat contoh bagaimana vektor eigen dan nilai eigen suatu matriks dihitung untuk lebih memahami metode ini:<\/p>\n<h2 class=\"wp-block-heading\"> Contoh penghitungan nilai eigen dan vektor eigen suatu matriks:<\/h2>\n<ul>\n<li> Temukan nilai eigen dan vektor eigen dari matriks berikut:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e82dbe4f6e975e1374cab2c1b74638b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}1&amp;0\\\\[1.1ex] 5&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Pertama, kita perlu mencari persamaan karakteristik matriks tersebut. Dan untuk itu determinan berikut harus diselesaikan:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-283812fe5eed97f58568fb6e515e3ff5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}1- \\lambda &amp;0\\\\[1.1ex] 5&amp;2-\\lambda \\end{vmatrix} = \\lambda^2-3\\lambda +2\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"338\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita menghitung akar-akar polinomial karakteristik, oleh karena itu, kita menyamakan hasil yang diperoleh dengan 0 dan menyelesaikan 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-2287061273d7f8502e0dbf1cb2fe1ad7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda^2-3\\lambda +2 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fdee5858b8b0187078ea372d9362900f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda= \\cfrac{-(-3)\\pm \\sqrt{(-3)^2-4\\cdot 1 \\cdot 2}}{2\\cdot 1} = \\cfrac{+3\\pm 1}{2}=\\begin{cases} \\lambda = 1 \\\\[2ex] \\lambda = 2 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"419\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Solusi persamaan tersebut adalah nilai eigen matriks.<\/p>\n<p> Setelah kita mendapatkan nilai eigen, kita menghitung vektor eigennya. Untuk melakukan ini, kita perlu menyelesaikan sistem berikut untuk setiap nilai eigen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-372f1009cb2b47f939cf9291f0f23885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-\\lambda I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Pertama-tama kita akan menghitung vektor eigen yang terkait dengan nilai eigen 1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-372f1009cb2b47f939cf9291f0f23885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-\\lambda I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f0cbd7a7e0670410881dcc0bfd4969c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-1 I)\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"163\" 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-e1f49b7ecec643964e4a14cd17ddecb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}0&amp;0\\\\[1.1ex] 5&amp;1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"156\" 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-06473aeaa487551bca2eb98ff786c8f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 0x+0y = 0 \\\\[2ex] 5x+y = 0\\end{array}\\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"112\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dari persamaan ini kita memperoleh subruang 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-82da1eb92338b5dc67c9e65188b6c247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle y=-5x\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"66\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Subruang vektor eigen juga disebut ruang eigen.<\/p>\n<p> Sekarang kita harus mencari basis dari ruang bersih ini, jadi kita berikan misalnya nilai 1 pada variabelnya<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> dan kami memperoleh vektor eigen 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-f4700ed7bb632b97f0ce1bec12409888_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle x = 1 \\ \\longrightarrow \\ y=-5\\cdot 1 = -5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"216\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8af03064a8f197990df832e71472cab0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] -5\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"79\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Terakhir, setelah vektor eigen yang terkait dengan nilai eigen 1 ditemukan, kita ulangi proses menghitung vektor eigen untuk nilai eigen 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-372f1009cb2b47f939cf9291f0f23885_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-\\lambda I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3d52ccfc2cbc996d3844af6c699a81b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-2I)\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"163\" 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-24442a53901cc9f0622aecf66ef2dc25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-1&amp;0\\\\[1.1ex] 5&amp;0\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"169\" 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-fbd3a434bf3f89ed38a893a98befee97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -x+0y = 0 \\\\[2ex] 5x+0y = 0\\end{array}\\right\\} \\longrightarrow \\ x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"207\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Dalam hal ini, hanya komponen pertama dari vektor yang harus bernilai 0, sehingga kita dapat memberikan nilai apa pun<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> . Namun untuk lebih mudahnya lebih baik diberi angka 1:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f47b6a21a448d003d909c0c1c969b8f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"66\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kesimpulannya, nilai eigen dan vektor eigen dari matriks 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-1668ed5f36ad0a8fcb28a264c76b6163_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 1 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] -5 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"158\" 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-56b0287c0bea71a1e5a258373aaa47d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"144\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Setelah Anda mengetahui cara mencari nilai eigen dan vektor eigen suatu matriks, Anda mungkin bertanya-tanya\u2026 dan untuk apa? Ternyata mereka sangat berguna untuk <a href=\"https:\/\/mathority.org\/id\/cara-mendiagonalisasi-matriks-yang-dapat-didiagonalisasi-latihan-diagonalisasi-matriks-2x2-3x3-4x4-diselesaikan-langkah-demi-langkah\/\">diagonalisasi matriks<\/a> , bahkan itu adalah aplikasi utamanya. Untuk mempelajari lebih lanjut, kami sarankan untuk memeriksa cara mendiagonalisasi matriks dengan tautan, yang prosedurnya dijelaskan langkah demi langkah dan terdapat juga contoh serta latihan yang diselesaikan untuk dipraktikkan.<\/p>\n<h2 class=\"wp-block-heading\"> Latihan soal nilai eigen dan vektor eigen (nilai eigen dan vektor eigen)<\/h2>\n<h3 class=\"wp-block-heading\"> Latihan 1<\/h3>\n<p> Hitung nilai eigen dan vektor eigen matriks persegi orde 2 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-0c6e3869ea2848140f026afc2ff8d554_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}3&amp;1\\\\[1.1ex] 2&amp;4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita hitung determinan matriks dikurangi \u03bb pada diagonal utamanya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fadce42062bb04b7477318fdc35c4285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}3- \\lambda &amp;1\\\\[1.1ex] 2&amp;4-\\lambda \\end{vmatrix} = \\lambda^2-7\\lambda +10\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"348\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita hitung akar-akar polinomial karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7139127430fa6b78b78715d57a6fdf1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda^2-7\\lambda +10=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda = 2 \\\\[2ex] \\lambda = 5 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"239\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 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-1b024c5f7e5acd0be55824c37befc587_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A- 2I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-614f9247b0d79635f70ec79eaa8c6529_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}1&amp;1\\\\[1.1ex] 2&amp;2\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"156\" 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-272495fa6e8f89ba4e7c6a6d848cb38a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+y = 0 \\\\[2ex] 2x+2y = 0\\end{array}\\right\\} \\longrightarrow \\ x=-y\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-77c240aaa8b75f1e5353c295ee86ad50_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"79\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita menghitung vektor eigen yang terkait dengan nilai eigen 5: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f48052a078660236820e9f605996e193_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-5I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8aa7cae3057d78343128cd1095df24e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-2&amp;1\\\\[1.1ex] 2&amp;-1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"183\" 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-d8c38e500cf7103b1dc0e91ea1b4531a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2x+y = 0 \\\\[2ex] 2x-y = 0\\end{array}\\right\\} \\longrightarrow \\ y=2x\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8be56f81b5aef28783636f85c4dbd643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"66\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, nilai eigen dan vektor eigen matriks A 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-cde889d89562f2e42bd6610b0045c118_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] -1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"158\" 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-3e954ba60fdc7eba60ba8530980854c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 5 \\qquad v = \\begin{pmatrix}1\\\\[1.1ex] 2 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"144\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Latihan 2<\/h3>\n<p> Tentukan nilai eigen dan vektor eigen matriks persegi 2&#215;2 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-54b0188c9fbadd6c3e35315443b71efd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;1\\\\[1.1ex] 3&amp;0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"95\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Kita hitung terlebih dahulu determinan matriks dikurangi \u03bb pada diagonal utamanya untuk memperoleh persamaan karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-88fcd3b21ad2fa5a4d1d7789a86043e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2- \\lambda &amp;1\\\\[1.1ex] 3&amp;-\\lambda \\end{vmatrix} = \\lambda^2-2\\lambda -3\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"323\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang mari kita hitung akar-akar polinomial karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2614817b28bdb25c4fd89d4c773b4e35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda^2-2\\lambda -3=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda = -1 \\\\[2ex] \\lambda = 3 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen -1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3e4bb6ba47bb4b9084b8a34d03dd35f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-(-1)I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"135\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdddf8c6da3e8066da62f60da7e9c603_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A+1I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e4b3d926f1a25454c3e645d79b28887d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 3&amp;1\\\\[1.1ex] 3&amp;1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"156\" 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-a7a6529e3ed8eb1607caa88475bcbb8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 3x+1y = 0 \\\\[2ex] 3x+1y = 0\\end{array}\\right\\} \\longrightarrow \\ y=-3x\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"225\" 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-67716508d5a9772f98c3f006f012dff1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] -3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"79\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita menghitung vektor eigen yang terkait dengan nilai eigen 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-50e802072a0f6e2942bc873d6a466909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-3I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-20e5d0be7e6dbe91bf15c835dac63b38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix}-1&amp;1\\\\[1.1ex] 3&amp;-3\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\end{pmatrix} =}\\begin{pmatrix}0 \\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"183\" 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-8355b1ade79ba1508633f309926bc221_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -1x+1y = 0 \\\\[2ex] 3x-3y = 0\\end{array}\\right\\} \\longrightarrow \\ y=x\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0f3cac5769795f1730fcbf118fdfbbc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"66\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, nilai eigen dan vektor eigen matriks A 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-95e2b0bf0405bc0c301600cbb4b2b28a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = -1 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] -3 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"172\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6322d97c5d24c1227b06dddf4b0974c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 3 \\qquad v = \\begin{pmatrix}1\\\\[1.1ex] 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"144\" 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 3<\/h3>\n<p> Tentukan nilai eigen dan vektor eigen matriks orde 3 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a0e4f4147cbc9e0b657ff432f64bc8e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}1&amp;2&amp;0\\\\[1.1ex] 2&amp;1&amp;0\\\\[1.1ex] 0&amp;1&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita harus menyelesaikan determinan matriks A dikurangi matriks identitas dikalikan lambda untuk memperoleh persamaan karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af2ff4694103925883916b6a974c84d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}1-\\lambda&amp;2&amp;0\\\\[1.1ex] 2&amp;1-\\lambda&amp;0\\\\[1.1ex] 0&amp;1&amp;2-\\lambda\\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"281\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini, kolom terakhir determinan memiliki dua angka nol, jadi kita akan memanfaatkan kolom ini untuk menghitung determinan berdasarkan kofaktor (atau komplemen) melalui kolom 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-ec7e7a2ec96b8d0721392c28838d105e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix}1-\\lambda&amp;2&amp;0\\\\[1.1ex] 2&amp;1-\\lambda&amp;0\\\\[1.1ex] 0&amp;1&amp;2-\\lambda\\end{vmatrix}&amp; = (2-\\lambda)\\cdot  \\begin{vmatrix}1-\\lambda&amp;2\\\\[1.1ex] 2&amp;1-\\lambda \\end{vmatrix} \\\\[3ex] &amp; = (2-\\lambda)[\\lambda^2 -2\\lambda -3] \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"136\" width=\"364\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita perlu menghitung akar-akar polinomial karakteristik. Sebaiknya jangan mengalikan tanda kurung karena dengan begitu kita akan memperoleh polinomial derajat ketiga, sebaliknya jika kedua faktor diselesaikan secara terpisah akan lebih mudah untuk mendapatkan nilai eigennya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-adbfb1815d4a480c0584dfee1d8039fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (2-\\lambda)[\\lambda^2 -2\\lambda -3]=0 \\ \\longrightarrow \\ \\begin{cases} 2-\\lambda=0 \\ \\longrightarrow \\ \\lambda = 2 \\\\[2ex] \\lambda^2 -2\\lambda -3=0 \\ \\longrightarrow \\begin{cases}\\lambda = -1 \\\\[2ex] \\lambda = 3 \\end{cases} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"489\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 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-6c6944f71d79a33d4789affbc82db4c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-2I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a6a12c460df4d2f44709c4fd595193dc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -1&amp;2&amp;0\\\\[1.1ex] 2&amp;-1&amp;0\\\\[1.1ex] 0&amp;1&amp;0\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c24e4b7b060a826203e3a049ddfc191_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -x+2y = 0 \\\\[2ex] 2x-y = 0\\\\[2ex] y=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=0 \\\\[2ex] x=y=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"248\" 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-75ebf6f61121b67afd80cdcec30a1709_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen -1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-76cc8bb12c3b49d4964b2b3f661677ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A+I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d23c4438a53032df27cc5334d4437c18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2&amp;2&amp;0\\\\[1.1ex] 2&amp;2&amp;0\\\\[1.1ex] 0&amp;1&amp;3\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"188\" 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-e023d57d34510e5e8f3a37c20d170e72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2x+2y = 0 \\\\[2ex] 2x+2y = 0\\\\[2ex] y+3z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=-y \\\\[2ex] y=-3z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"233\" 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-0be9ef18fb17845818bdd9de51dcb114_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}3 \\\\[1.1ex] -3 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 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-50e802072a0f6e2942bc873d6a466909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-3I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d0360154b87545dd87e1b0b7bc06f4e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -2&amp;2&amp;0\\\\[1.1ex] 2&amp;-2&amp;0\\\\[1.1ex] 0&amp;1&amp;-1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"229\" 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-b8d514791286e43eae4b09d893d528df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2x+2y = 0 \\\\[2ex] 2x-2y = 0\\\\[2ex] y-z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=y \\\\[2ex] y=z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"224\" 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-99f86f65a5a9c69119285377d88f2efa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, nilai eigen dan vektor eigen matriks A 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-fe42249314c1698847242c608bd65843_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"146\" 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-d412e1f81df9d6425db73113aaae5cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = -1 \\qquad v = \\begin{pmatrix}3 \\\\[1.1ex] -3 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"174\" 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-f581aa37c9698dfb32062777a5a75b11_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 3 \\qquad v = \\begin{pmatrix}1\\\\[1.1ex] 1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"146\" 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 4<\/h3>\n<p> Hitung nilai eigen dan vektor eigen matriks persegi 3&#215;3 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1323184f42d56f070e5b46a75a2e5c4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;1&amp;3\\\\[1.1ex]-1&amp;1&amp;1\\\\[1.1ex] 1&amp;2&amp;4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita selesaikan determinan matriks dikurangi \u03bb pada diagonal utamanya untuk mendapatkan persamaan karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bc48c8489b25004ef131cc6ced36b929_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2-\\lambda&amp;1&amp;3\\\\[1.1ex]-1&amp;1-\\lambda&amp;1\\\\[1.1ex] 1&amp;2&amp;4-\\lambda\\end{vmatrix}=-\\lambda^3+7\\lambda^2-10\\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"437\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami mengekstrak faktor persekutuan dari polinomial karakteristik dan menyelesaikan \u03bb dari setiap 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-411dab2f65b426c37f8427d81ef13e97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda(-\\lambda^2+7\\lambda-10)=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda=0\\\\[2ex] -\\lambda^2+7\\lambda-10=0 \\ \\longrightarrow \\begin{cases}\\lambda = 2 \\\\[2ex] \\lambda = 5 \\end{cases} \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"481\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 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-0e3b04137690f84b723e3ed568e1114a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-0I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bd7ebe2424c6524d522d5bba16d72d33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2&amp;1&amp;3\\\\[1.1ex]-1&amp;1&amp;1\\\\[1.1ex] 1&amp;2&amp;4\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"202\" 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-12cd10d2dc8afdb7a045beae4946b64d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2x+y+3z= 0 \\\\[2ex] -x+y+z= 0\\\\[2ex] x+2y+4z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=-\\cfrac{2z}{3} \\\\[4ex] y=-\\cfrac{5z}{3} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"104\" width=\"266\" 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-a34877b285f281c83d7e73fa8eb40b9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}-2 \\\\[1.1ex] -5\\\\[1.1ex] 3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 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-6c6944f71d79a33d4789affbc82db4c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-2I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fbcc697a3be877838fae3507dd3c1b68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 0&amp;1&amp;3\\\\[1.1ex]-1&amp;-1&amp;1\\\\[1.1ex] 1&amp;2&amp;2\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bb8d470bc7bff9f5d8d5a0245b1e7cbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} y+3z = 0 \\\\[2ex] -x-y+z= 0\\\\[2ex] x+2y+2z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=-3z \\\\[2ex] x=4z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"263\" 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-30e589c5ae6b940b901454c296d8342b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}4\\\\[1.1ex] -3 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 5: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f48052a078660236820e9f605996e193_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-5I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aaf6f17dedf5eecd1e035b9da59da2c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -3&amp;1&amp;3\\\\[1.1ex]-1&amp;-4&amp;1\\\\[1.1ex] 1&amp;2&amp;-1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"229\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\">\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dca8569528fb4923639dd535e25a0f74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -3x+y+3z = 0 \\\\[2ex] -x-4y+z = 0\\\\[2ex] x+2y-z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=z \\\\[2ex] y=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"255\" 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-308b2f0f597fcc084d8d06d6c45fd3e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 0 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, nilai eigen dan vektor eigen matriks A 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-62d8a98f007b72910fcd79622eda19e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 0 \\qquad v = \\begin{pmatrix}-2 \\\\[1.1ex] -5 \\\\[1.1ex] 3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"160\" 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-ee67e876a46b09430d2d73a653f2d743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}4 \\\\[1.1ex] -3 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"160\" 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-99e4e8b0b837c26991777a294f30d49a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 5 \\qquad v = \\begin{pmatrix}1\\\\[1.1ex] 0 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"146\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<h3 class=\"wp-block-heading\"> Latihan 5<\/h3>\n<p> Hitung nilai eigen dan vektor eigen matriks 3&#215;3 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a39253beac54a05e9e84d431daf43362_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix}2&amp;2&amp;2\\\\[1.1ex] 1&amp;2&amp;0\\\\[1.1ex] 0&amp;1&amp;3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"122\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita selesaikan determinan matriks dikurangi \u03bb pada diagonal utamanya untuk mendapatkan persamaan karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9392bbf957bee6c445c64192ae96a2ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}2-\\lambda&amp;2&amp;2\\\\[1.1ex] 1&amp;2-\\lambda&amp;0\\\\[1.1ex] 0&amp;1&amp;3-\\lambda\\end{vmatrix}=-\\lambda^3+7\\lambda^2-14\\lambda+8\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"468\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kita mencari akar polinomial karakteristik atau polinomial minimum menggunakan aturan Ruffini:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-152ec29207fec8bdac7dabe9e1fbff31_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{r|rrrr} &amp; -1&amp;7&amp;-14&amp;8 \\\\[2ex] 1 &amp; &amp; -1&amp;6&amp;-8 \\\\ \\hline &amp;-1\\vphantom{\\Bigl)}&amp;6&amp;-8&amp;0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"190\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dan kemudian kita menemukan akar polinomial yang diperoleh:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b92304d107c097ec5712527929011440_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -\\lambda^2+6\\lambda -8=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda =2 \\\\[2ex] \\lambda = 4 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Jadi nilai eigen matriksnya 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-05492792022e885b332adb0cbba45a0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda=1 \\qquad \\lambda =2 \\qquad \\lambda = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"200\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-78173073be8dbdcab8a122ade043906d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-1I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-981cc7881e44436326a35a7cc36ad26a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1&amp;2&amp;2\\\\[1.1ex] 1&amp;1&amp;0\\\\[1.1ex] 0&amp;1&amp;2\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"188\" 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-d928870722dec65e8b48f7175d5dd4ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} x+2y+2z= 0 \\\\[2ex] x+y= 0\\\\[2ex] y+2z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=-y \\\\[2ex] y=-2z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"263\" 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-da5ca9263773369d5824688b71a31644_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}2 \\\\[1.1ex] -2\\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 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-6c6944f71d79a33d4789affbc82db4c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-2I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b9d1686e2947a9bbe1dc10b373128e1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 0&amp;2&amp;2\\\\[1.1ex] 1&amp;0&amp;0\\\\[1.1ex] 0&amp;1&amp;1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"188\" 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-6000063fd1cc954e119cd5d73d08c405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2y+2z = 0 \\\\[2ex] x= 0\\\\[2ex] y+z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=-z \\\\[2ex] x=0\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"222\" 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-d47216be7fc08447ac3022a105a086b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0\\\\[1.1ex] -1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 4: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0545c0847763140ccc62a58cf4207c6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-4I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-15e38e9899a9e8bb47cfbf10a4f05075_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -2&amp;2&amp;2\\\\[1.1ex] 1&amp;-2&amp;0\\\\[1.1ex] 0&amp;1&amp;-1\\end{pmatrix}\\begin{pmatrix}x \\\\[1.1ex] y \\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"229\" 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-004d61132ba8eeee123d8614432cbce2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2x+2y+2z = 0 \\\\[2ex] x-2y = 0\\\\[2ex] y-z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} x=2y \\\\[2ex] y=z \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"273\" 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-5871bb6e88776aab87e0239540d43677_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}2 \\\\[1.1ex] 1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, nilai eigen dan vektor eigen matriks A 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-f1ea8e2eff0c179b9872da8f6fab2d4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 1 \\qquad v = \\begin{pmatrix}2\\\\[1.1ex] -2 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"160\" 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-edc6fd09f9c6a12b26518a9103cc6610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 2 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] -1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"160\" 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-b492e98d771e76e77dc68d2fe2ea92c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 4 \\qquad v = \\begin{pmatrix}2 \\\\[1.1ex] 1 \\\\[1.1ex] 1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"146\" 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 6<\/h3>\n<p> Tentukan nilai eigen dan vektor eigen dari matriks 4\u00d74 berikut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5cb04190d6f536d33b22265317441144_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix}1&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-1&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;2&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"189\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E3F2FD boto_ver_solucion\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E3F2FD\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>lihat solusi<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Pertama-tama kita harus menyelesaikan determinan matriks dikurangi \u03bb pada diagonal utamanya untuk mendapatkan persamaan karakteristik:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-35cae2dd143d77e22a522b49e8d43f3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A-\\lambda I)= \\begin{vmatrix}1-\\lambda&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-1-\\lambda&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;2-\\lambda&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;3-\\lambda\\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"108\" width=\"352\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dalam hal ini, kolom terakhir determinan hanya berisi nol kecuali satu elemen, oleh karena itu kita akan memanfaatkan kolom ini untuk menghitung determinan berdasarkan kofaktor melalui kolom 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-456b0612b308c03fd1643a5ba0f332e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{aligned} \\begin{vmatrix}1-\\lambda&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-1-\\lambda&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;2-\\lambda&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;3-\\lambda\\end{vmatrix}&amp; = (3-\\lambda)\\cdot  \\begin{vmatrix}1-\\lambda&amp;0&amp;-1\\\\[1.1ex] 2&amp;-1-\\lambda&amp;-3\\\\[1.1ex] -2&amp;0&amp;2-\\lambda\\end{vmatrix} \\\\[3ex] &amp; = (3-\\lambda)[-\\lambda^3 +2\\lambda^2 +3\\lambda] \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"161\" width=\"505\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Sekarang kita harus menghitung akar-akar polinomial karakteristik. Sebaiknya jangan mengalikan tanda kurung karena dengan begitu kita akan memperoleh polinomial derajat keempat, sebaliknya jika kedua faktor diselesaikan secara terpisah akan lebih mudah untuk menghitung nilai eigennya: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef6e59f8631cac087c988004aa512b62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (3-\\lambda)[-\\lambda^3 +2\\lambda^2 +3\\lambda]=0 \\ \\longrightarrow \\ \\begin{cases} 3-\\lambda=0 \\ \\longrightarrow \\ \\lambda = 3 \\\\[2ex] -\\lambda^3 +2\\lambda^2 +3\\lambda =0 \\ \\longrightarrow \\ \\lambda(-\\lambda^2 +2\\lambda +3) =0 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"620\" 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-786b2892e7045f117498697407d35552_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda(-\\lambda^2 +2\\lambda +3)=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda=0  \\\\[2ex] -\\lambda^2 +2\\lambda +3=0 \\ \\longrightarrow \\ \\begin{cases} \\lambda=-1 \\\\[2ex] \\lambda = 3 \\end{cases}\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"483\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 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-0e3b04137690f84b723e3ed568e1114a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-0I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f43f22947b29779ef456e4ac7a5d66a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 1&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-1&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;2&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;3\\end{pmatrix}\\begin{pmatrix}w \\\\[1.1ex] x \\\\[1.1ex] y\\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"258\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c1d7e96203dceb7288f89ab932532351_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} w-y = 0 \\\\[2ex] 2w-x-3y = 0\\\\[2ex] -2w+2y=0 \\\\[2ex] 3z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} w=y \\\\[2ex] x=-w  \\\\[2ex]z=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"263\" 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-3c4a8b3ef3502a2bf8efd6cc398b5ae6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] -1 \\\\[1.1ex] 1  \\\\[1.1ex]0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen -1: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cdddf8c6da3e8066da62f60da7e9c603_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A+1I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7fbbdceca419f15672da0dcb7c15078c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} 2&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;0&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;3&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;4\\end{pmatrix}\\begin{pmatrix}w \\\\[1.1ex] x \\\\[1.1ex] y\\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"244\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3ff9a5afa0cefa73985e7ba00c945dac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} 2w-y = 0 \\\\[2ex] 2w-3y = 0\\\\[2ex] -2w+3y=0 \\\\[2ex] 4z=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=w=0  \\\\[2ex]z=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"263\" 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-daee73fdcebacce8a5e5f7104ed9c213_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\\\[1.1ex] 0  \\\\[1.1ex]0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Kami menghitung vektor eigen yang terkait dengan nilai eigen 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-50e802072a0f6e2942bc873d6a466909_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A-3I)v=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6493c6019a8b9be3254db2ffeaa19703_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} -2&amp;0&amp;-1&amp;0\\\\[1.1ex] 2&amp;-4&amp;-3&amp;0\\\\[1.1ex] -2&amp;0&amp;-1&amp;0\\\\[1.1ex] 0&amp;0&amp;0&amp;0\\end{pmatrix}\\begin{pmatrix}w \\\\[1.1ex] x \\\\[1.1ex] y\\\\[1.1ex] z \\end{pmatrix} =\\begin{pmatrix}0 \\\\[1.1ex] 0\\\\[1.1ex] 0 \\\\[1.1ex] 0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"258\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fbcbd01420d80be317ecbec57010b662_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\left.\\begin{array}{l} -2w-y = 0 \\\\[2ex] 2w-4x-3y = 0\\\\[2ex] -2w-y=0 \\\\[2ex] 0=0 \\end{array}\\right\\} \\longrightarrow \\ \\begin{array}{l} y=-2w \\\\[2ex] x=2w  \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"129\" width=\"280\" 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-3b70eeb51bea073f058763401adf5240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}1 \\\\[1.1ex] 2 \\\\[1.1ex] -2  \\\\[1.1ex]0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Nilai eigen 3 mempunyai multiplisitas sama dengan 2, karena diulang dua kali. Oleh karena itu kita harus mencari vektor eigen lain yang memenuhi persamaan yang sama:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5dc5fd38503b7683d8a7e3df9da9ee8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle v = \\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0  \\\\[1.1ex]1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"68\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Oleh karena itu, nilai eigen dan vektor eigen matriks A 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-b8bd7188d1d3ed1abe178d9b5f5bbc0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 0 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] -1 \\\\[1.1ex] 1  \\\\[1.1ex]0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"160\" 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-2ea128d2a6e5387bd538ac3d0119b2ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = -1 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] 1 \\\\[1.1ex] 0  \\\\[1.1ex]0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"160\" 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-22d83a8f13bdb44bf1c23f3c6b963d65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 3 \\qquad v = \\begin{pmatrix}1 \\\\[1.1ex] 2 \\\\[1.1ex] -2  \\\\[1.1ex]0\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"160\" 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-65cd815fe71a1c6d8063f0f78e3422a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lambda = 3 \\qquad v = \\begin{pmatrix}0 \\\\[1.1ex] 0 \\\\[1.1ex] 0  \\\\[1.1ex]1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"146\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Pada halaman ini kami menjelaskan apa itu nilai eigen dan vektor eigen, disebut juga nilai eigen dan vektor eigen. Anda juga akan menemukan contoh cara menghitungnya serta latihan langkah demi langkah untuk dipraktikkan. Apa yang dimaksud dengan nilai eigen dan vektor eigen? Walaupun pengertian nilai eigen dan vektor eigen sulit dipahami, namun definisinya adalah sebagai &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\"> <span class=\"screen-reader-text\">Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks<\/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-331","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>Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks -<\/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\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks -\" \/>\n<meta property=\"og:description\" content=\"Pada halaman ini kami menjelaskan apa itu nilai eigen dan vektor eigen, disebut juga nilai eigen dan vektor eigen. Anda juga akan menemukan contoh cara menghitungnya serta latihan langkah demi langkah untuk dipraktikkan. Apa yang dimaksud dengan nilai eigen dan vektor eigen? Walaupun pengertian nilai eigen dan vektor eigen sulit dipahami, namun definisinya adalah sebagai &hellip; Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks Selengkapnya &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T06:35:36+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-710a5e2df8739c35c060f790f5592734_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=\"1 menit\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\"},\"author\":{\"name\":\"Tim Mathority\",\"@id\":\"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38\"},\"headline\":\"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks\",\"datePublished\":\"2023-07-06T06:35:36+00:00\",\"dateModified\":\"2023-07-06T06:35:36+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\"},\"wordCount\":177,\"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\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\",\"url\":\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\",\"name\":\"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks -\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/id\/#website\"},\"datePublished\":\"2023-07-06T06:35:36+00:00\",\"dateModified\":\"2023-07-06T06:35:36+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/#breadcrumb\"},\"inLanguage\":\"id\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/id\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks\"}]},{\"@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":"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks -","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\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/","og_locale":"id_ID","og_type":"article","og_title":"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks -","og_description":"Pada halaman ini kami menjelaskan apa itu nilai eigen dan vektor eigen, disebut juga nilai eigen dan vektor eigen. Anda juga akan menemukan contoh cara menghitungnya serta latihan langkah demi langkah untuk dipraktikkan. Apa yang dimaksud dengan nilai eigen dan vektor eigen? Walaupun pengertian nilai eigen dan vektor eigen sulit dipahami, namun definisinya adalah sebagai &hellip; Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/","article_published_time":"2023-07-06T06:35:36+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-710a5e2df8739c35c060f790f5592734_l3.png"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"1 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks","datePublished":"2023-07-06T06:35:36+00:00","dateModified":"2023-07-06T06:35:36+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/"},"wordCount":177,"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\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/","url":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/","name":"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks -","isPartOf":{"@id":"https:\/\/mathority.org\/id\/#website"},"datePublished":"2023-07-06T06:35:36+00:00","dateModified":"2023-07-06T06:35:36+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/#breadcrumb"},"inLanguage":"id","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/id\/menghitung-nilai-eigennilai-eigen-dan-vektor-eigenvektor-eigen-suatu-matriks\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/id\/"},{"@type":"ListItem","position":2,"name":"Nilai eigen (atau nilai eigen) dan vektor eigen (atau vektor eigen) dari suatu matriks"}]},{"@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\/331","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=331"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts\/331\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/media?parent=331"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/categories?post=331"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/tags?post=331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}