{"id":311778,"date":"2023-01-31T10:30:30","date_gmt":"2023-01-31T03:30:30","guid":{"rendered":"https:\/\/quipperhome.wpcomstaging.com\/?p=311778"},"modified":"2023-02-06T10:30:03","modified_gmt":"2023-02-06T03:30:03","slug":"transpose-matriks","status":"publish","type":"post","link":"https:\/\/quipperhome.wpcomstaging.com\/mapel\/matematika\/transpose-matriks\/","title":{"rendered":"Pengertian Transpose Matriks, Sifat dan Bentuk Penulisannya"},"content":{"rendered":"\n
\"\"<\/figure>\n\n\n\n

Hai Quipperian, bagaimana kabarnya? Semoga selalu sehat dan tetap semangat, ya.<\/p>\n\n\n\n

Di pembahasan sebelumnya, Quipper Blog sudah membahas tentang invers matriks, kan? Untuk menentukan invers matriks, kamu harus tahu dulu determinan dan adjoin matriks tersebut. Nah,<\/em> adjoin merupakan istilah lain untuk menyatakan transpose matriks. Lalu, apa yang dimaksud transpose matriks? Yuk, artikel selengkapnya berikut ini.<\/p>\n\n\n\n

Pengertian Transpose Matriks<\/strong><\/h2>\n\n\n\n

Transpose matriks adalah matriks baru yang elemen baris dan kolomnya merupakan elemen kolom dan baris matriks sebelumnya. Artinya, transpose matriks dibentuk oleh pembalikan elemen baris menjadi kolom dan elemen kolom menjadi baris. Jika matriks yang akan dijadikan transpose bukan matriks persegi, maka ordo pada transposenya merupakan kebalikan dari ordo matriks sebelumnya. Misalnya, matriks ordo 2 x 3 memiliki transpose matriks yang ordonya 3 x 2, matriks 3 x 1 memiliki transpose matriks yang ordonya 1 x 3, dan seterusnya. Namun, jika bentuknya matriks persegi, transpose matriksnya tetap, misal matriks 2 x 2 memiliki transpose matriks 2 x 2, matriks 3 x 3 memiliki transpose matriks 3 x 3, dan seterusnya.<\/p>\n\n\n\n

Bentuk Penulisan Transpose Matriks<\/strong><\/h2>\n\n\n\n

Bentuk penulisan transpose matriks sama dengan matriks asalnya. Hanya saja, ada tambahan pangkat T pada nama matriksnya. Misalnya matriks awalnya P, maka transpose matriksnya PT<\/sup>. Adapun contoh transpose matriks adalah sebagai berikut.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Matriks P di atas merupakan matriks yang memiliki ordo 3 x 2. Untuk mengubahnya menjadi transpose matriks, kamu harus mengubah letak elemen baris menjadi elemen kolom dan elemen kolom menjadi baris seperti berikut.<\/p>\n\n\n\n

\"\"\/<\/figure>\n\n\n\n
\"\"\/<\/figure>\n\n\n\n

Dari contoh di atas, terlihat kan, jika elemen baris berubah menjadi elemen kolom dan elemen kolom berubah menjadi elemen baris? <\/p>\n\n\n\n

Matriks P (matriks awal) memiliki ordo 3 x 2 dan transpose matriksnya memiliki ordo 2 x 3.<\/p>\n\n\n\n

Di antara semua besaran yang dipelajari terkait matriks, besaran transpose ini termasuk yang paling mudah. Kamu hanya perlu menukar elemen-elemen baris dan kolomnya.<\/p>\n\n\n\n

\n

Baca Juga: <\/span> Langkah Mencari Invers Matriks 2x2Lengkap dengan Contoh Soal dan Pembahasan<\/a><\/p>\n<\/div>\n\n\n\n

Sifat-Sifat Transpose Matriks<\/strong><\/h2>\n\n\n\n

Adapun sifat-sifat yang dimiliki oleh transpose matriks adalah sebagai berikut.<\/p>\n\n\n\n

    \n
  1. (PT<\/sup>)T<\/sup> = P<\/li>\n<\/ol>\n\n\n\n

    Jika suatu transpose matriks ditransposekan, maka akan dihasilkan matriks awal. Untuk membuktikannya, coba transposekan matriks PT<\/sup> pada contoh sebelumnya.<\/sub><\/p>\n\n\n\n

    Dari contoh di atas, terlihat bahwa transpose matriks PT<\/sup> yang ditransposekan akan sama dengan matriks P awalnya.<\/p>\n\n\n\n

      \n
    1. (P + Q)T<\/sup> = PT<\/sup> + QT<\/sup><\/li>\n<\/ol>\n\n\n\n

      Sifat kedua ini berlaku untuk transpose penjumlahan dua buah matriks. Ingat, dua matriks atau lebih bisa dijumlahkan dengan syarat ordo keduanya harus sama. Jika penjumlahan dua matriks yang ordonya sama ditransposekan, hasilnya akan sama dengan penjumlahan masing-masing transpose matriks. Perhatikan contoh berikut.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Jika dijumlahkan, akan menjadi:<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Selanjutnya, transposekan matriks P + Q di atas.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Sekarang, coba kamu transposekan dahulu masing-masing matriksnya.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Lalu, jumlahkan kedua transpose tersebut.<\/p>\n\n\n\n

      \"\"<\/figure>\n\n\n\n

      Dari kedua perhitungan, diperoleh hasil yang sama kan? Jika menjumpai soal demikian, kamu bisa memilih cara yang dianggap paling mudah dan cepat.<\/p>\n\n\n\n

        \n
      1. (P – Q)T<\/sup> = PT<\/sup> – QT<\/sup><\/li>\n<\/ol>\n\n\n\n

        Sifat ketiga ini berlaku untuk transpose pengurangan dua buah matriks. Ingat, dua matriks atau lebih bisa dikurangkan dengan syarat ordo keduanya harus sama. Jika pengurangan dua matriks yang ordonya sama ditransposekan, hasilnya akan sama dengan pengurangan masing-masing transpose matriks. Namun, pada pengurangan dua matriks tidak bisa dibalik atau tidak berlaku sifat komutatif, ya. Perhatikan contoh berikut.<\/p>\n\n\n\n

        \"\"<\/figure>\n\n\n\n

        Jika dijumlahkan, akan menjadi:<\/p>\n\n\n\n

        \"\"<\/figure>\n\n\n\n

        Selanjutnya, transposekan matriks P – Q di atas.<\/p>\n\n\n\n

        \"\"<\/figure>\n\n\n\n

        Sekarang, coba kamu transposekan dahulu masing-masing matriksnya.<\/p>\n\n\n\n

        \"\"<\/figure>\n\n\n\n

        Lalu, jumlahkan kedua transpose tersebut.<\/p>\n\n\n\n

        \"\"<\/figure>\n\n\n\n

        Dari kedua perhitungan, diperoleh hasil yang sama kan? Jika menjumpai soal demikian, kamu bisa memilih cara yang dianggap paling mudah dan cepat.<\/p>\n\n\n\n

          \n
        1. (K<\/em>P)T<\/sup> = k.<\/em>PT<\/sup>, di mana k<\/em> = suatu konstanta<\/li>\n<\/ol>\n\n\n\n

          Jika suatu konstanta atau skalar dikalikan dengan matriks, lalu matriks hasil perkalian tersebut ditansposekan, maka hasilnya akan sama dengan perkalian transpose matriks dengan konstanta tersebut. Perhatikan contoh berikut.<\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          Perkalian antara matriks dan k<\/em> akan menghasilkan:<\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          Lalu, transposekan matriks di atas.<\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          Sekarang, coba transposekan dahulu matriksnya, lalu kalikan hasilnya dengan k<\/em> = 3.<\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          Selanjutnya, kalikan transpose di atas dengan k<\/em> = 3.<\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          Ternyata, hasilnya sama kan? Mudah bukan transpose matriks itu?<\/p>\n\n\n\n

            \n
          1. (PQ)T<\/sup> = PT<\/sup>QT<\/sup><\/li>\n<\/ol>\n\n\n\n

            Sifat terakhir ini, berkaitan dengan perkalian antarmatriks. Apakah kamu ingat syarat perkalian antarmatriks? Dua matriks bisa dikalikan dengan syarat jumlah kolom matriks pertama sama dengan jumlah baris matriks kedua. Contoh matriks 2 x 3 bisa dikalikan dengan matriks 3 x 1. Lalu, bagaimana dengan matriks persegi? Syarat untuk matriks persegi sama dengan matriks-matriks lainnya, yaitu jumlah kolom matriks pertama harus sama dengan jumlah baris matriks kedua. Jika kedua matriks dikalikan lalu ditransposekan, maka hasilnya akan sama dengan hasil perkalian antartranspose matriksnya.<\/p>\n\n\n\n

            Contoh Soal Transpose Matriks<\/strong><\/h2>\n\n\n\n

            Untuk mengasah pemahamanmu, yuk simak contoh soal berikut ini.<\/p>\n\n\n\n

            Contoh Soal 1<\/strong><\/h3>\n\n\n\n

            Tentukan transpose dari matriks-matriks berikut.<\/p>\n\n\n\n

              \n
            1. <\/sub><\/li>\n\n\n\n
            2. <\/sub><\/li>\n\n\n\n
            3. <\/sub><\/li>\n<\/ol>\n\n\n\n

              Pembahasan:<\/p>\n\n\n\n

              Untuk menentukan transpose, semua elemen baris diubah menjadi elemen kolom. Sama seperti pembahasan sebelumnya. Adapun transpose dari matriks pada poin a \u2013 c adalah sebagai berikut.<\/p>\n\n\n\n

                \n
              1. <\/sub><\/li>\n\n\n\n
              2. <\/sub><\/li>\n\n\n\n
              3. <\/sub><\/li>\n<\/ol>\n\n\n\n

                Contoh Soal 2<\/strong><\/h3>\n\n\n\n

                Diketahui suatu transpose matriks berikut ini.<\/p>\n\n\n\n

                \"\"<\/figure>\n\n\n\n

                Jika matriks P dijumlahkan dengan matriks Q = 0 1 -3 5 , akan dihasilkan matriks baru P + Q = 4 6 -7 8 . Tentukan nilai a<\/em> + b<\/em>!<\/p>\n\n\n\n

                Pembahasan:<\/p>\n\n\n\n

                Diketahui:<\/p>\n\n\n\n

                \"\"<\/figure>\n\n\n\n

                Q = 0 1 -3 5<\/p>\n\n\n\n

                P + Q = 4 6 -7 8<\/p>\n\n\n\n

                Ditanya: a<\/em> + b<\/em> =\u2026?<\/p>\n\n\n\n

                Jawab:<\/p>\n\n\n\n

                Mula-mula, tentukan dahulu matriks P.<\/p>\n\n\n\n

                \"\"<\/figure>\n\n\n\n

                Selanjutnya, jumlahkan matriks P dan matrik Q hingga dihasilkan matriks P + Q.<\/p>\n\n\n\n

                \"\"<\/figure>\n\n\n\n

                Dari perhitungan di atas, diperoleh:<\/p>\n\n\n\n

                a<\/em> = 4<\/p>\n\n\n\n

                b<\/em> + 5 = 8 \u2194 b<\/em> = 3<\/p>\n\n\n\n

                Jadi, nilai a<\/em> + b<\/em> = 4 + 3 = 7.<\/p>\n\n\n\n

                Contoh Soal 3<\/strong><\/h3>\n\n\n\n

                Diketahui dua matriks seperti berikut.<\/p>\n\n\n\n

                \"\"<\/figure>\n\n\n\n

                Tentukan hasil perkalian transpose matriks M dan N!<\/p>\n\n\n\n

                Pembahasan:<\/p>\n\n\n\n

                Untuk menyelesaikan soal ini, kamu bisa lihat kembali sifat transpose matriks nomor 5, yaitu (PQ)T<\/sup> = PT<\/sup>QT<\/sup>. Untuk memudahkanmu, kalikan dahulu matriksnya lalu transposekan.<\/p>\n\n\n\n

                \"\"<\/figure>\n\n\n\n

                Lalu, transposekan hasilnya.<\/p>\n\n\n\n

                \"\"<\/figure>\n\n\n\n

                Jadi, hasil transposenya sama dengan hasil perkalian kedua matriks itu sendiri.<\/p>\n\n\n\n

                Itulah pembahasan Quipper Blog kali ini. Semoga bermanfaat, ya. Untuk mendapatkan materi lengkapnya, yuk buruan gabung Quipper Video. Salam Quipper!<\/p>\n","protected":false},"excerpt":{"rendered":"

                Hai Quipperian, bagaimana kabarnya? Semoga selalu sehat dan tetap semangat, ya. Di pembahasan sebelumnya, Quipper Blog sudah membahas tentang invers matriks, kan? Untuk menentukan…<\/p>\n","protected":false},"author":156447303,"featured_media":311782,"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":[679384865],"tags":[679384825],"ppma_author":[679386823,679386836],"class_list":["post-311778","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-matematika","tag-materi-matematika-kelas-11"],"acf":[],"yoast_head":"\nPengertian Transpose Matriks, Sifat dan Bentuk Penulisannya - Quipper Blog<\/title>\n<meta name=\"description\" content=\"Transpose matriks adalah matriks baru yang elemen baris dan kolomnya merupakan elemen kolom dan baris matriks sebelumnya.. 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