{"id":267591,"date":"2019-07-22T17:38:30","date_gmt":"2019-07-22T10:38:30","guid":{"rendered":"https:\/\/www.quipper.com\/id\/blog\/?p=267591"},"modified":"2022-03-01T19:26:43","modified_gmt":"2022-03-01T12:26:43","slug":"matriks-dan-contoh-soalnya","status":"publish","type":"post","link":"https:\/\/quipperhome.wpcomstaging.com\/mapel\/matematika\/matriks-dan-contoh-soalnya\/","title":{"rendered":"Kelas 12, Yuk Belajar Matriks dan Contoh Soalnya di Sini!"},"content":{"rendered":"<p><img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-267831\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/Kelas-12-YukBelajar-Matriks-dan-Contoh-Soalnya-di-Sini1-2.png\" alt=\"\" width=\"800\" height=\"533\" \/><\/p>\n<blockquote><p>Matriks merupakan kumpulan bilangan, simbol, atau ekspresi dengan bentuk persegi panjang, sesuai baris dan kolom. Ordo matriks menunjukkan banyaknya baris dan kolom dalam matriks. Transpose matriks adalah bentuk operasi matriks di mana susunan baris diubah jadi kolom dan kolom diubah jadi baris. Matriks sama adalah matriks yang punya ordo-ordo yang sama dan tiap elemennya terletak pada baris dan kolom yang sama.<\/p><\/blockquote>\n<p>Hai Quipperian, bagaimana kabarnya? Semoga selalu sehat dan tetap semangat belajar, ya!<br \/>\nApa jadinya dunia ini tanpa ilmu pengetahuan? Pernahkah Quipperian berpikir demikian? Dunia tanpa ilmu pengetahuan ibarat rumah tanpa lampu. Berkembangnya ilmu pengetahun tidak lepas dari peran para ilmuwan. Mereka rela menghabiskan waktu, tenaga, pikiran hanya untuk mengamati, memahami, dan menganalisis gejala alam di sekitar. Hal itu tentu tidak mudah, mengingat banyak variabel yang harus mereka pertimbangkan sebelum merumuskan ilmu pengetahuan yang baku dan bisa diterima semua orang.<br \/>\nSaat menghadapi permasalahan dengan berbagai variabel, metode yang biasa digunakan adalah matriks. Pada artikel kali ini, Quipper Blog akan mengajak Quipperian untuk belajar tentang matriks dan contoh soalnya. Yuk, simak!<\/p>\n<h2><strong>Pengertian Matriks<\/strong><\/h2>\n<p><img decoding=\"async\" class=\"alignnone size-full wp-image-267841\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/Kelas-12-YukBelajar-Matriks-dan-Contoh-Soalnya-di-Sini-2.png\" alt=\"\" width=\"800\" height=\"533\" \/><br \/>\nMatriks adalah kumpulan bilangan, simbol, atau ekspresi yang disusun membentuk persegi panjang berdasarkan baris dan kolom. Notasi matriks dinyatakan dalam huruf kapital. Berikut ini contoh penulisan matriks.<br \/>\n<img decoding=\"async\" class=\"alignnone size-full wp-image-267741\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-1-2.png\" alt=\"\" width=\"205\" height=\"158\" \/><br \/>\nMatriks D di atas memiliki sembilan elemen, yaitu <em>a<\/em>&#8211;<em>i.<\/em>Letak elemen dinyatakan dalam fungsi <em>x<sub>p,q<\/sub><\/em>di mana <em>p<\/em>menunjukkan baris, sedangkan <em>q<\/em>menunjukkan kolom, contohnya <em>x<sub>2,3<\/sub><\/em>= <em>f<\/em>; <em>x<sub>3,1<\/sub><\/em>=<em>g<\/em>; dan <em>x<sub>2,2<\/sub><\/em>= <em>e<\/em>. Baris merupakan bagian matriks yang mengarah horizontal, sedangkan kolom merupakan bagian matriks yang mengarah vertikal.<\/p>\n<h2><strong>Ordo Matriks<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267821\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/Kelas-12-YukBelajar-Matriks-dan-Contoh-Soalnya-di-Sini2-2.png\" alt=\"\" width=\"800\" height=\"533\" \/><br \/>\nOrdo atau ukuran matriks menunjukkan banyaknya baris dan kolom di dalam matriks. Ordo biasa dinotasikan sebagai \u00a0\u2211 baris x kolom.\u00a0Perhatikan contoh berikut.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267731\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-2-2.png\" alt=\"\" width=\"404\" height=\"489\" \/><br \/>\nBagaimana Quipperian, sudah paham tentang ordo matriks, belum? Jika sudah, selanjutnya Quipper Blog akan membahas tentang transpose matriks.<\/p>\n<h2><strong>Transpose Matriks<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267811\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/Kelas-12-YukBelajar-Matriks-dan-Contoh-Soalnya-di-Sini3-2.png\" alt=\"\" width=\"800\" height=\"533\" \/><br \/>\nTranspose matriks merupakan bentuk operasi matriks di mana susunan baris diubah menjadi kolom, sedangkan bagian kolom diubah menjadi baris. Baris ke-<em>p<\/em>diubah menjadi kolom ke-<em>p<\/em>atau kolom ke-<em>q<\/em>diubah menjadi baris ke-<em>q<\/em>. Jika matriks D di atas dijadikan transpose matriks D, notasi yang digunakan adalah D<sup>T.\u00a0<\/sup>Untuk lebih jelasnya, perhatikan contoh berikut.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267721\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-3-2.png\" alt=\"\" width=\"394\" height=\"174\" \/><br \/>\nBerdasarkan transpose matriks di atas, Quipperian bisa melihat bahwa elemen baris ke-1, yaitu 1, 2, 3, dituliskan pada kolom ke-1, elemen baris ke-2, yaitu 4, 5, 6, dituliskan pada kolom ke-2, dan begitu seterusnya. Dengan demikian, transpose matriks bisa mengubah ordo matriks jika jumlah baris dan kolomnya tidak sama.<\/p>\n<h2><strong>Matriks Sama<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267801\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/Kelas-12-YukBelajar-Matriks-dan-Contoh-Soalnya-di-Sini4-2.png\" alt=\"\" width=\"800\" height=\"533\" \/><br \/>\nSuatu matriks dikatakan sama jika matriks-matriks tersebut memiliki ordo yang sama dan setiap elemennya terletak pada baris dan kolom yang sama. Jika suatu matriks sama, otomatis setiap elemen yang seletak nilainya sama. Perhatikan contoh soal berikut.<\/p>\n<h2><strong>Contoh Soal 1<\/strong><\/h2>\n<p>Diketahui dua matriks sebagai berikut.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267711\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-4-2.png\" alt=\"\" width=\"505\" height=\"121\" \/><br \/>\nTentukan nilai dari <em>a<\/em>, <em>b<\/em>,<em>c<\/em>, dan <em>d<\/em>!<br \/>\n<strong>Pembahasan:<\/strong><br \/>\nUntuk mencari nilai <em>a<\/em>, <em>b<\/em>,<em>c<\/em>, dan <em>d<\/em>, Quipperian harus tahu bahwa matriks D = E, sehingga elemen seletak nilainya pasti sama.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267701\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-5-2.png\" alt=\"\" width=\"332\" height=\"195\" \/><br \/>\nDari perhitungan di atas, diperoleh nilai <em>a<\/em>= 1, <em>b <\/em>= 6, <em>c<\/em>= 0, dan <em>d<\/em>= 9.<br \/>\nJadi, nilai <em>a<\/em>, <em>b<\/em>,<em>c<\/em>, dan <em>d<\/em>berturut-turut adalah 1, 6, 0, dan 9.<\/p>\n<h2><strong>Operasi Antarmatriks<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267791\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/Kelas-12-YukBelajar-Matriks-dan-Contoh-Soalnya-di-Sini5-2.png\" alt=\"\" width=\"800\" height=\"533\" \/><br \/>\nLayaknya bilangan, matriks juga bisa dioperasikan seperti penjumlahan, pengurangan, perkalian angka dengan matriks, maupun perkalian antarmatriks.<\/p>\n<h3><strong>1. Penjumlahan dan pengurangan matriks<\/strong><\/h3>\n<p>Penjumlahan dan pengurangan matriks hanya bisa dilakukan jika ordonya sama, misalnya matriks 2 \u00d7 2 dikurangkan dengan matriks 2 \u00d7 2 lainnya. Elemen yang dijumlahkan atau dikurangkan harus seletak, artinya posisi atau letaknya sama. Perhatikan contoh berikut.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267691\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-6-2.png\" alt=\"\" width=\"401\" height=\"437\" \/><br \/>\nBerdasarkan contoh di atas, terlihat bahwa penjumlahan atau pengurangan matriks tidak mengakibatkan perubahan ordo.<\/p>\n<h3><strong>2. Perkalian angka dengan matriks<\/strong><\/h3>\n<p>Semua matriks bisa dikalikan dengan konstanta atau bilangan berapapun. Jika dikalikan dengan suatu konstanta atau bilangan, semua elemen di dalam matriks tersebut harus dikalikan satu per satu dengan konstanta yang dimaksud. Contohnya sebagai berikut.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267681\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-7-2.png\" alt=\"\" width=\"278\" height=\"227\" \/><br \/>\nBerdasarkan hasil di atas, ternyata perkalian antara konstanta dan matriks tidak akan mengubah ordo matriks tersebut.<\/p>\n<h3><strong>3. Perkalian antara matriks dan matriks<\/strong><\/h3>\n<p>Jika dibandingkan operasi matriks sebelumnya, perkalian antara matriks dan matriks ini terbilang lebih rumit. Untuk mengalikan antara matriks dan matriks, Quipperian harus mengalikan seluruh elemen tiap baris ke-<em>p<\/em>dengan kolom ke-<em>\u00adp<\/em>, lalu hasilnya dijumlahkan pada baris yang sama. Misalnya diketahui perkalian matriks sebagai berikut.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267671\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-8-2.png\" alt=\"\" width=\"290\" height=\"149\" \/><br \/>\nContoh mengoperasikan perkalian antara dua matriks di atas adalah sebagai berikut.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267661\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-9-2.png\" alt=\"\" width=\"252\" height=\"302\" \/><br \/>\n<strong>Pembahasan:<\/strong><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267651\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-10-2.png\" alt=\"\" width=\"849\" height=\"451\" \/><br \/>\nHal yang harus diingat dari perkalian matriks adalah sifat perkalian matriks tidak berlaku bolak-balik atau \u00a0AB\u00a0\u2260 BA.<br \/>\nAgar Quipperian semakin paham dengan pembahasan matriks ini, perhatikan contoh soal berikut.<\/p>\n<h2><strong>Contoh Soal 2<\/strong><\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267641\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-11-2.png\" alt=\"\" width=\"994\" height=\"157\" \/><br \/>\n<strong>Pembahasan:<\/strong><br \/>\nSebelum menyelesaikan soal di atas, Quipperian jabarkan kembali persamaannya, yaitu sebagai berikut.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267631\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-12-2.png\" alt=\"\" width=\"463\" height=\"373\" \/><br \/>\nSelanjutnya, tentukan nilai <em>x<\/em>dan <em>y<\/em>berdasarkan nilai elemen seletak.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267621\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-13-2.png\" alt=\"\" width=\"285\" height=\"138\" \/><br \/>\nDiperoleh nilai <em>x<\/em>= 2 dan <em>y<\/em>= 4. Dengan demikian, <em>x<\/em>+ <em>2xy<\/em>+ <em>y<\/em>= 2 + 2(2)(4) + 4 = 22.<br \/>\nJadi, nilai <em>x<\/em>+ <em>2xy<\/em>+ <em>y<\/em>= 22.<\/p>\n<h2><strong>Contoh soal 3<\/strong><\/h2>\n<p>Tentukan nilai 2<em>a<\/em><sup>2<\/sup>+ <em>b<\/em>\u2013 <em>c<\/em>\u00a0 yang memenuhi persamaan matriks berikut.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267611\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-14-2.png\" alt=\"\" width=\"429\" height=\"107\" \/><br \/>\n<strong>Pembahasan:<\/strong><br \/>\nUntuk menentukan nilai 2<em>a<\/em><sup>2<\/sup>+ <em>b<\/em>\u2013 <em>c<\/em>, Quipperian harus mengalikan matriks-matriks di sisi kiri terlebih dahulu.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267601\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-15-2.png\" alt=\"\" width=\"440\" height=\"279\" \/><br \/>\nDari persamaan di atas, diperoleh:<br \/>\nBaris ke-2, kolom ke-2<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267771\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-16-2.png\" alt=\"\" width=\"208\" height=\"56\" \/><br \/>\nBaris ke-1, kolom ke-2<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267761\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-17-2.png\" alt=\"\" width=\"167\" height=\"183\" \/><br \/>\nBaris ke-1, kolom ke-1<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-267751\" src=\"https:\/\/quipperhome.wpcomstaging.com\/wp-content\/uploads\/2019\/07\/matriks-18-2.png\" alt=\"\" width=\"167\" height=\"176\" \/><br \/>\nDengan demikian, nilai 2<em>a<\/em><sup>2<\/sup>+ <em>b<\/em>\u2013 <em>c<\/em>= 2(3)<sup>2<\/sup>+ (-3) \u2013 1 = 14.<br \/>\nJadi, nilai 2<em>a<\/em><sup>2<\/sup>+ <em>b<\/em>\u2013 <em>c<\/em>= 14.<br \/>\nItulah bahasan tentang matriks dan contoh soalnya. Pada prinsipnya, belajar matriks tidaklah sulit karena semua sudah ada aturannya masing-masing, bagaimana harus menjumlahkan, mengurangkan, dan mengalikan. Jangan lupa gabung dengan <strong><a href=\"https:\/\/learn.quipper.com\/en\/login\">Quipper Video<\/a><\/strong>agar Quipperian bisa <em>update<\/em>materi-materi terbaru, baik tentang matriks maupun materi yang lain. Salam Quipper!<br \/>\n<strong>Sumber:<\/strong><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Penulis: Eka Viandari<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Matriks merupakan kumpulan bilangan, simbol, atau ekspresi dengan bentuk persegi panjang, sesuai baris dan kolom. Ordo matriks menunjukkan banyaknya baris dan kolom dalam matriks.&hellip;<\/p>\n","protected":false},"author":99400369,"featured_media":267831,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_crdt_document":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[679384888,679384865],"tags":[],"ppma_author":[679386827],"class_list":["post-267591","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mapel","category-matematika"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Kelas 12, Yuk Belajar Matriks dan Contoh Soalnya di Sini!<\/title>\n<meta name=\"description\" content=\"Belajar Matematika paling asyik itu mengulas tentang matriks dan contoh soalnya, Quipperian! 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