{"id":333,"date":"2023-07-06T05:26:21","date_gmt":"2023-07-06T05:26:21","guid":{"rendered":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/"},"modified":"2023-07-06T05:26:21","modified_gmt":"2023-07-06T05:26:21","slug":"contoh-matriks-ortogonal-sifat-2x2-3x3","status":"publish","type":"post","link":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/","title":{"rendered":"Matriks ortogonal"},"content":{"rendered":"<p>Di halaman ini Anda akan melihat apa itu matriks ortogonal dan hubungannya dengan invers suatu matriks. Anda juga akan melihat beberapa contoh untuk memahaminya dengan sempurna. Selain itu, kami mengajari Anda rumus yang memeriksa matriks ortogonal apa pun, yang dengannya Anda akan mengetahui cara menemukannya dengan cepat. Dan terakhir, Anda akan menemukan properti dan penerapan matriks khusus ini serta latihan ujian yang diselesaikan secara umum.<\/p>\n<h2 class=\"wp-block-heading\"> Apa itu matriks ortogonal?<\/h2>\n<p> Pengertian matriks ortogonal 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>Matriks ortogonal<\/strong> adalah matriks bilangan real persegi yang dikalikan transposnya (atau transposnya) sama dengan matriks identitas. Artinya, kondisi berikut terpenuhi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ce7debd9ea0083703f398f280e534f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A\\cdot A^t = A^t \\cdot A =I\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"147\" 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 ortogonal dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-afd3cedfe0f405ed9f2d585b5ac1d8cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: 0px;\"><\/p>\n<p> mewakili matriks yang ditransposisikan.<\/p>\n<\/div>\n<p> Agar kondisi ini terpenuhi, kolom dan baris matriks ortogonal harus merupakan vektor satuan ortogonal, yaitu harus membentuk basis ortonormal. Oleh karena itu, beberapa ahli matematika juga menyebutnya <strong>matriks ortonormal<\/strong> .<\/p>\n<h2 class=\"wp-block-heading\"> Kebalikan dari matriks ortogonal<\/h2>\n<p> Cara lain untuk menjelaskan konsep matriks ortogonal adalah melalui matriks invers, karena <strong>transpos (atau transpos) matriks suatu matriks ortogonal sama dengan inversnya.<\/strong><\/p>\n<p> Untuk memahami teorema ini sepenuhnya, penting bagi Anda untuk mengetahui cara <a href=\"https:\/\/mathority.org\/id\/matriks-terbalik\/\">membalikkan matriks<\/a> . Di tautan ini Anda akan menemukan penjelasan mendetail tentang invers matriks, semua propertinya, dan Anda bahkan memiliki latihan penyelesaian langkah demi langkah untuk dipraktikkan.<\/p>\n<p> Matriks invers suatu matriks ortogonal dapat dengan mudah ditunjukkan ekuivalennya dengan transposnya menggunakan kondisi matriks ortogonal dan sifat utama matriks invers:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-36f7666e4730a6311c088c7e8d7f0f38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{c} A \\cdot A^t =I \\\\[2ex] A \\cdot A^{-1} = I\\end{array} \\right\\} \\longrightarrow \\ A^t=A^{-1}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"231\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, matriks ortogonal akan selalu merupakan <a href=\"https:\/\/mathority.org\/id\/kapan-contoh-dan-sifat-matriks-beraturan-atau-matriks-inversi\/\">matriks yang dapat dibalik<\/a> , atau dengan kata lain matriks beraturan atau matriks tak berdegenerasi.<\/p>\n<p> Selanjutnya kita akan melihat beberapa contoh matriks ortogonal untuk melengkapi pemahaman konsep segala sesuatu.<\/p>\n<h2 class=\"wp-block-heading\"> Contoh matriks ortogonal 2\u00d72<\/h2>\n<p> Matriks berikut merupakan matriks ortogonal berdimensi 2\u00d72: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/matrice-orthogonale-de-dimension-22152-1.webp\" alt=\"matriks ortogonal berdimensi 2x2\" class=\"wp-image-3302\" width=\"132\" height=\"70\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Kita dapat memeriksa ortogonalnya dengan menghitung hasil kali berdasarkan transposnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d157361ae2a13dbeabc4ba1aab7f8a94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"44\" 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-f7baa091c2fd963507b93e6bec5c386b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t= \\begin{pmatrix} 0 &amp; 1 \\\\[1.1ex] -1 &amp; 0 \\end{pmatrix} \\cdot \\begin{pmatrix} 0 &amp; -1 \\\\[1.1ex] 1 &amp; 0 \\end{pmatrix} = \\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"315\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Karena hasilnya menghasilkan matriks Identik, kita verifikasi bahwa A adalah matriks ortogonal.<\/p>\n<h2 class=\"wp-block-heading\"> Contoh matriks ortogonal 3\u00d73<\/h2>\n<p> Matriks berikut merupakan matriks ortogonal berdimensi 3\u00d73: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/matrice-orthogonale-de-dimension-32153-1.webp\" alt=\"matriks ortogonal berdimensi 3x3\" class=\"wp-image-3304\" width=\"193\" height=\"102\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Kita dapat menunjukkan bahwa matriks tersebut ortogonal dengan mengalikan matriks A dengan transposnya:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-35687f56ff4ad5d1b19ea673b4ac85de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t = \\begin{pmatrix}0.8&amp;0.6&amp;0\\\\[1.1ex] -0.6&amp;0.8&amp;0\\\\[1.1ex] 0&amp;0&amp;1\\end{pmatrix}\\cdot \\begin{pmatrix}0.8&amp;-0.6&amp;0\\\\[1.1ex] 0.6&amp;0.8&amp;0\\\\[1.1ex] 0&amp;0&amp;1\\end{pmatrix}= \\begin{pmatrix} 1 &amp; 0 &amp; 0\\\\[1.1ex] 0 &amp; 1 &amp; 0 \\\\[1.1ex] 0&amp;0&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"453\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Karena penyelesaiannya adalah matriks kesatuan, kita tunjukkan bahwa A adalah matriks ortogonal.<\/p>\n<h2 class=\"wp-block-heading\"> Rumus mencari matriks ortogonal 2&#215;2<\/h2>\n<div class=\"adsb30\" style=\" margin:px; text-align:\"><\/div>\n<p> Kita kemudian akan melihat bukti bahwa semua matriks ortogonal orde 2 mengikuti pola yang sama.<\/p>\n<p> Pertimbangkan matriks generik berukuran 2\u00d72:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ac19d6ab63d390a9340cbce4014b1136_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\begin{pmatrix} a &amp; b \\\\[1.1ex] c &amp; d \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"96\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Agar matriks ini ortogonal, persamaan matriks berikut harus dipenuhi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05ba7bc31dc95f239c8ddb0ffdd72a81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t =I\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" 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-1e108701513ef6f2118e3b7d32657cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} a &amp; b \\\\[1.1ex] c &amp; d \\end{pmatrix} \\cdot \\begin{pmatrix} a &amp; c \\\\[1.1ex] b &amp; d \\end{pmatrix} =\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"216\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Menyelesaikan perkalian matriks, kita memperoleh persamaan 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-d5435c614cb0da442fe04f65aec89637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{pmatrix} a^2+b^2 &amp; ac+bd \\\\[1.1ex] ac+bd &amp; c^2+d^2 \\end{pmatrix}=\\begin{pmatrix} 1 &amp; 0 \\\\[1.1ex] 0 &amp; 1 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" 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-f8897132ecdbf389450e8c5fa1707226_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{c}a^2+b^2=1 \\\\[2ex] ac+bd=0 \\\\[2ex] c^2+d^2=1 \\end{array} \\qquad \\begin{array}{l} (1) \\\\[2ex] (2) \\\\[2ex] (3) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"162\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Jika Anda perhatikan lebih dekat, persamaan ini sangat mirip dengan <em>hubungan dasar trigonometri Pythagoras<\/em> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cbd8ab83a790807844d1d30e63429337_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\sin ^2\\alpha+\\cos ^2\\alpha=1\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Oleh karena itu, suku-suku yang memenuhi persamaan (1) dan (3) yang diperoleh 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-9abeb023c5050d8d7f6fbab8c52227ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{array}{l} a = \\cos \\theta \\qquad \\qquad \\qquad c = \\sin\\phi \\\\[2ex] b = \\sin \\theta \\qquad \\qquad \\qquad d = \\cos \\phi\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"242\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Selain itu, dengan mensubstitusikan nilai-nilai tersebut ke dalam persamaan kedua, kita memperoleh hubungan antara kedua sudut: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9b210cbf7eb8602c723c54204fc5ad8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle ac+bd=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"88\" 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-8c5edfeb3556ee37b43da4afaeb0c3f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos\\theta\\sin\\phi+\\sin\\theta\\cos\\phi=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"202\" 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-b2b1e1085946911044e2758ca2783eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\tan\\phi=-\\tan\\theta\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"117\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Artinya, salah satu dari dua syarat berikut harus dipenuhi:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0eb6746afdcb971294de82ecebad37b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{si} \\quad c=\\sin\\phi=-\\sin\\theta \\quad \\longrightarrow \\quad  d=\\cos\\phi=\\cos\\theta\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"373\" 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-6face2f33e95163135f12204424969f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{si} \\quad d=\\cos \\phi=-\\cos \\theta \\quad \\longrightarrow \\quad c=\\sin\\phi=\\sin\\theta\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"373\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Jadi, kesimpulannya, matriks ortogonal harus mempunyai struktur salah satu dari dua matriks berikut: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-la-matrice-orthogonale-de-dimension-22152-1.webp\" alt=\"rumus matriks ortogonal berdimensi 2x2\" class=\"wp-image-3267\" width=\"623\" height=\"143\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> 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-356a08e839ab6974a16448e16e56745d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\theta\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> adalah bilangan real.<\/p>\n<p> Memang kalau sebagai contoh kita memberi nilai<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d3f94c7fef174aa94efe99c9aa192cab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\theta=\\frac{\\pi}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"45\" style=\"vertical-align: -12px;\"><\/p>\n<p> dan kita ambil struktur pertama, kita akan memperoleh matriks yang telah kita verifikasi ortogonalnya pada bagian \u201cContoh matriks ortogonal 2\u00d72\u201d: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1a331cab64745933f7c8a5009c799be6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle M_1 \\left(\\theta =\\frac{\\pi}{2}\\right)=\\begin{pmatrix} \\cos \\cfrac{\\pi}{2} &amp;\\sin \\cfrac{\\pi}{2} \\\\[4ex] -\\sin \\cfrac{\\pi}{2} &amp; \\cos \\cfrac{\\pi}{2} \\end{pmatrix}=\\begin{pmatrix} \\vphantom{\\frac{\\pi}{2}}0 &amp;1 \\\\[2ex]\\vphantom{\\frac{\\pi}{2}} -1 &amp; 0 \\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"366\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-118\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"> Properti Matriks Ortogonal<\/h2>\n<p> Ciri-ciri matriks jenis ini adalah:<\/p>\n<ul>\n<li> Matriks ortogonal tidak akan pernah menjadi <a href=\"https:\/\/mathority.org\/id\/matriks-tunggal-atau-degenerasi\/\">matriks tunggal<\/a> karena matriks tersebut selalu dapat dibalik. Dalam pengertian ini, kebalikan dari matriks ortogonal adalah matriks ortogonal lainnya.<\/li>\n<\/ul>\n<ul>\n<li> Matriks ortogonal apa pun dapat didiagonalisasi. Kita kemudian mengatakan bahwa matriks ortogonal <em>dapat didiagonalisasi secara ortogonal.<\/em><\/li>\n<\/ul>\n<ul>\n<li> Semua nilai eigen atau nilai eigen suatu matriks ortogonal mempunyai modulus sama dengan 1.<\/li>\n<\/ul>\n<ul>\n<li> Matriks ortogonal apa pun yang hanya terdiri dari bilangan real juga merupakan matriks normal.<\/li>\n<\/ul>\n<ul>\n<li> Analog dari matriks ortogonal dalam lingkungan dengan bilangan kompleks adalah matriks kesatuan.<\/li>\n<\/ul>\n<ul>\n<li> Jelasnya matriks identitas merupakan matriks ortogonal.<\/li>\n<\/ul>\n<ul>\n<li> Himpunan matriks ortogonal berdimensi n \u00d7 n serta operasi hasil kali matriksnya membentuk suatu grup yang disebut grup ortogonal. Artinya, hasil kali dua matriks ortogonal sama dengan matriks ortogonal lainnya.<\/li>\n<\/ul>\n<ul>\n<li> Selain itu, hasil perkalian matriks ortogonal dengan transposnya dapat dinyatakan dengan delta Kronecker:<\/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-7d0922008f857f33f46de7551a8ff7cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (A\\cdot A^{t})_{ij} = \\delta_{ij}=\\begin{cases}1 &amp; \\mbox{si }i = j, \\\\[2ex] 0 &amp; \\mbox{si }i \\ne j\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"238\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Terakhir, determinan matriks ortogonal selalu +1 atau -1.<\/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-12d5717a2cb94708642478117c7c309d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{det}(A)=\\pm 1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Latihan matriks ortogonal yang diselesaikan<\/h2>\n<p> Selanjutnya kita akan menyelesaikan latihan matriks ortogonal.<\/p>\n<ul>\n<li> Diketahui matriks persegi berorde 3 berikut, tentukan nilai dari\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> Dan<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> untuk membuatnya ortogonal:<\/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-892ca58ec5cd36060396cb566902d65d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\frac{1}{3}\\begin{pmatrix}a&amp;a&amp;1\\\\[1.1ex] b&amp;1&amp;b\\\\[1.1ex] 1&amp;a&amp;a\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"140\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Agar ortogonalitas matriks terpenuhi, hasil kali matriks dengan transposnya harus sama dengan matriks Identitas. JADI:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-301d774ec2d0663c858c91e548000749_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A\\cdot A^t = I\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" 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-c2dc9ef8c514302f183ca66626cabc1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{3}\\begin{pmatrix}a&amp;a&amp;1\\\\[1.1ex] b&amp;1&amp;b\\\\[1.1ex] 1&amp;a&amp;a\\end{pmatrix} \\cdot \\frac{1}{3}\\begin{pmatrix}a&amp;b&amp;1\\\\[1.1ex] a&amp;1&amp;a\\\\[1.1ex] 1&amp;b&amp;a\\end{pmatrix}=\\begin{pmatrix}1&amp;0&amp;0\\\\[1.1ex] 0&amp;1&amp;0\\\\[1.1ex] 0&amp;0&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"334\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Kami mengalikan 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-18a21a22f3cc9747c310d271c3fe4c5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{9}\\begin{pmatrix}2a^2+1&amp;ab+a+b&amp;2a+a^2\\\\[1.5ex] ab+a+b&amp;2b^2+1&amp;b+a+ab\\\\[1.5ex] 2a+a^2&amp;b+a+ab&amp;1+2a^2\\end{pmatrix} =\\begin{pmatrix}1&amp;0&amp;0\\\\[1.5ex] 0&amp;1&amp;0\\\\[1.5ex] 0&amp;0&amp;1\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"87\" width=\"418\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Sekarang kita dapat memperoleh persamaan dari pojok kiri atas matriks, karena elemen-elemen pada posisi tersebut harus cocok. Belum: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a4d0f699410c3a7de5d6af181073f8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{9}(2a^2+1) = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"113\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-119\"><\/div>\n<\/div>\n<p> Kami memecahkan persamaan dan menghilangkan yang tidak diketahui: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c6700a19d58afe1d84668664734ef725_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2a^2+1 = 9\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"89\" 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-a2007d061c6ecfda4deef22882bfce17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2a^2 = 8\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" 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-5fd565f4a4ab39908601493ada1575cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle a^2 = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" 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-2f617bc4be3760ed1e13564672a4b6ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{a = \\pm 2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Namun, ada persamaan yang tidak berlaku untuk solusi positif, misalnya persamaan di pojok kanan atas. Jadi <strong>hanya solusi negatif yang mungkin dilakukan<\/strong> .<\/p>\n<p> Di sisi lain, untuk menghitung variabel<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> kita dapat mencocokkan, misalnya, suku-suku yang ditempatkan pada baris kedua kolom pertama:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16d30b554d24bd4b8a00b156ed1503d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{9}(ab+a+b) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"134\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fe062334bff68f2ba70f0f025d2a2d9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle ab+a+b = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"110\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Dengan mengganti nilai<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> dalam 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-e8e6825541c5d7e2a12df99d9dc7b3b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -2b-2+b = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" 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-a13af6b471ccdb2e6804fc02b87abc4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -b =2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-27e8de15a4a0f239878f4cc4f5b8db24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{b =-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"53\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Singkatnya, satu-satunya solusi yang mungkin 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-70dc1f4331f63bff4d12f4bad8ef34e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\bm{a=b =-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"86\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Jadi matriks ortogonal yang sesuai dengan nilai-nilai ini 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-cb7e7a27658da85f7b0d16b17f1f0815_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A=\\frac{1}{3}\\begin{pmatrix}-2&amp;-2&amp;1\\\\[1.1ex] -2&amp;1&amp;-2\\\\[1.1ex] 1&amp;-2&amp;-2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"179\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"> Penerapan matriks ortogonal<\/h2>\n<p> Walaupun kelihatannya tidak begitu karena biasanya bentuknya sangat sederhana, matriks ortogonal sangat penting dalam matematika, khususnya dalam bidang aljabar linier.<\/p>\n<p> Dalam geometri, matriks ortogonal mewakili transformasi isometrik (yang tidak mengubah jarak dan sudut) dalam ruang vektor nyata, oleh karena itu disebut transformasi ortogonal. Lebih lanjut, transformasi ini merupakan isomorfisme internal dari ruang vektor yang dipertimbangkan. Transformasi ini dapat berupa <strong>rotasi<\/strong> , <strong>refleksi specular<\/strong> , atau <strong>inversi<\/strong> .<\/p>\n<p> Terakhir, matriks jenis ini juga digunakan dalam fisika, karena memungkinkan mempelajari pergerakan benda tegar. Dan mereka bahkan digunakan dalam perumusan teori lapangan tertentu.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Di halaman ini Anda akan melihat apa itu matriks ortogonal dan hubungannya dengan invers suatu matriks. Anda juga akan melihat beberapa contoh untuk memahaminya dengan sempurna. Selain itu, kami mengajari Anda rumus yang memeriksa matriks ortogonal apa pun, yang dengannya Anda akan mengetahui cara menemukannya dengan cepat. Dan terakhir, Anda akan menemukan properti dan penerapan &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/\"> <span class=\"screen-reader-text\">Matriks ortogonal<\/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":[63],"tags":[],"class_list":["post-333","post","type-post","status-publish","format-standard","hentry","category-matriks-terbalik"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Matriks ortogonal - Mathoritas<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/\" \/>\n<meta property=\"og:locale\" content=\"id_ID\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Matriks ortogonal - Mathoritas\" \/>\n<meta property=\"og:description\" content=\"Di halaman ini Anda akan melihat apa itu matriks ortogonal dan hubungannya dengan invers suatu matriks. 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Mathoritas","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/","og_locale":"id_ID","og_type":"article","og_title":"Matriks ortogonal - Mathoritas","og_description":"Di halaman ini Anda akan melihat apa itu matriks ortogonal dan hubungannya dengan invers suatu matriks. Anda juga akan melihat beberapa contoh untuk memahaminya dengan sempurna. Selain itu, kami mengajari Anda rumus yang memeriksa matriks ortogonal apa pun, yang dengannya Anda akan mengetahui cara menemukannya dengan cepat. Dan terakhir, Anda akan menemukan properti dan penerapan &hellip; Matriks ortogonal Selengkapnya &raquo;","og_url":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/","article_published_time":"2023-07-06T05:26:21+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ce7debd9ea0083703f398f280e534f3_l3.png"}],"author":"Tim Mathority","twitter_card":"summary_large_image","twitter_misc":{"Ditulis oleh":"Tim Mathority","Estimasi waktu membaca":"4 menit"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/"},"author":{"name":"Tim Mathority","@id":"https:\/\/mathority.org\/id\/#\/schema\/person\/ea4523caf53a07e2ebf32e306a925b38"},"headline":"Matriks ortogonal","datePublished":"2023-07-06T05:26:21+00:00","dateModified":"2023-07-06T05:26:21+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/"},"wordCount":813,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/id\/#organization"},"articleSection":["Matriks terbalik"],"inLanguage":"id","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/","url":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/","name":"Matriks ortogonal - Mathoritas","isPartOf":{"@id":"https:\/\/mathority.org\/id\/#website"},"datePublished":"2023-07-06T05:26:21+00:00","dateModified":"2023-07-06T05:26:21+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/#breadcrumb"},"inLanguage":"id","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/id\/contoh-matriks-ortogonal-sifat-2x2-3x3\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/id\/"},{"@type":"ListItem","position":2,"name":"Matriks ortogonal"}]},{"@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\/333","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=333"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/posts\/333\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/media?parent=333"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/categories?post=333"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/id\/wp-json\/wp\/v2\/tags?post=333"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}