Roboguru

Determinan matriks K yang memenuhi persamaan (43​75​)⋅K=(32​11​) adalah ...

Pertanyaan

Determinan matriks K yang memenuhi persamaan open parentheses table row 4 7 row 3 5 end table close parentheses times K equals open parentheses table row 3 1 row 2 1 end table close parentheses adalah ...

  1. 3

  2. 1

  3. negative 1

  4. negative 2

  5. negative 3

Pembahasan Soal:

Persamaan matriks bentuk A times X equals B yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell A times X end cell equals B row X equals cell A to the power of negative 1 end exponent B end cell end table

Jika terdapat matriks A equals open parentheses table row a b row c d end table close parentheses, inversnya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row a b row c d end table close parentheses end cell row cell A to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator det space A end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator a times d minus b times c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses end cell end table

Jika terdapat matiks A equals open parentheses table row a b row c d end table close parentheses, determinannya yaitu;

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row a b row c d end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator a times d minus b times c end fraction end cell end table

Perhitungan pada soal, yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row 4 7 row 3 5 end table close parentheses times K end cell equals cell open parentheses table row 3 1 row 2 1 end table close parentheses end cell row K equals cell open parentheses table row 4 7 row 3 5 end table close parentheses to the power of negative 1 end exponent times open parentheses table row 3 1 row 2 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 4 open parentheses 5 close parentheses minus 7 open parentheses 3 close parentheses end fraction open parentheses table row 5 cell negative 7 end cell row cell negative 3 end cell 4 end table close parentheses times open parentheses table row 3 1 row 2 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 20 minus 21 end fraction open parentheses table row 5 cell negative 7 end cell row cell negative 3 end cell 4 end table close parentheses times open parentheses table row 3 1 row 2 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 1 end fraction open parentheses table row cell 5 open parentheses 3 close parentheses plus open parentheses negative 7 close parentheses open parentheses 2 close parentheses end cell cell 5 open parentheses 1 close parentheses plus open parentheses negative 7 close parentheses open parentheses 1 close parentheses end cell row cell negative 3 open parentheses 3 close parentheses plus 4 open parentheses 2 close parentheses end cell cell negative 3 open parentheses 1 close parentheses plus 4 open parentheses 1 close parentheses end cell end table close parentheses end cell row blank equals cell negative 1 open parentheses table row cell 15 minus 14 end cell cell 5 minus 7 end cell row cell negative 9 plus 8 end cell cell negative 3 plus 4 end cell end table close parentheses end cell row blank equals cell negative 1 open parentheses table row 1 cell negative 2 end cell row cell negative 1 end cell 1 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell 2 row 1 cell negative 1 end cell end table close parentheses end cell end table

Kemudian tentukan nilai determinan dari matriks K.

table attributes columnalign right center left columnspacing 0px end attributes row K equals cell open parentheses table row cell negative 1 end cell 2 row 1 cell negative 1 end cell end table close parentheses end cell row cell det space K end cell equals cell fraction numerator 1 over denominator negative 1 open parentheses negative 1 close parentheses minus 2 open parentheses 1 close parentheses end fraction end cell row blank equals cell fraction numerator 1 over denominator 1 minus 2 end fraction end cell row blank equals cell fraction numerator 1 over denominator negative 1 end fraction end cell row blank equals cell negative 1 end cell end table

Determinan matriks K yang memenuhi persamaan open parentheses table row 4 7 row 3 5 end table close parentheses times K equals open parentheses table row 3 1 row 2 1 end table close parentheses adalah negative 1.

Jadi, jawaban yang tepat adalah C.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Ayu

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Terakhir diupdate 07 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

Pertanyaan yang serupa

Diketahui matriks A=(2−2​3−4​) dan Bt=(−2−1​65​). Jika A−1 adalah invers matriks A, maka det(A−1B)= ....

Pembahasan Soal:

Jika diketahui matriks A equals open parentheses table row a b row c d end table close parentheses, maka dapat ditentukan determinan matriks sebagai berikut.

text det  end text A equals a d minus b c

Sifat determinan:

text det  end text A to the power of negative 1 end exponent equals fraction numerator 1 over denominator text det end text space A end fraction

text det end text space open parentheses A times B close parentheses equals text det end text space A times text det end text space B

Diketahui B to the power of t equals open parentheses table row cell negative 2 end cell 6 row cell negative 1 end cell 5 end table close parentheses sehingga diperoleh B equals open parentheses table row cell negative 2 end cell cell negative 1 end cell row 6 5 end table close parentheses 

Berdasarkan sifat determinan matriks diperoleh hubungan berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell text det end text open parentheses A to the power of negative 1 end exponent B close parentheses end cell equals cell text det end text open parentheses A to the power of negative 1 end exponent close parentheses times text det  end text B end cell row blank equals cell fraction numerator 1 over denominator text det end text space A end fraction times text det  end text B end cell row blank equals cell fraction numerator 1 over denominator 2 times open parentheses negative 4 close parentheses minus 3 times open parentheses negative 2 close parentheses end fraction times open parentheses negative 2 times 5 minus open parentheses negative 1 close parentheses times 6 close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 2 end fraction open parentheses negative 4 close parentheses end cell row blank equals 2 end table

Oleh karena itu, jawaban yang tepat adalah C.

0

Roboguru

Diketahui matriks A=(15​−1−4​),B=(72​31​). Jika A−1 invers matriks A dan B−1 adalah invers matriks B, maka deteminan matriks B−1A−1 adalah ...

Pembahasan Soal:

Diketahui:

A equals open parentheses table row 1 cell negative 1 end cell row 5 cell negative 4 end cell end table close parentheses comma thin space B equals open parentheses table row 7 3 row 2 1 end table close parentheses

Mencari invers matriks A:

table attributes columnalign right center left columnspacing 0px end attributes row cell A to the power of negative 1 end exponent end cell equals cell open parentheses table row 1 cell negative 1 end cell row 5 cell negative 4 end cell end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell fraction numerator 1 over denominator det open parentheses table row 1 cell negative 1 end cell row 5 cell negative 4 end cell end table close parentheses end fraction open parentheses table row cell negative 4 end cell cell negative open parentheses negative 1 close parentheses end cell row cell negative 5 end cell 1 end table close parentheses end cell row blank equals cell 1 over 1 open parentheses table row cell negative 4 end cell cell negative open parentheses negative 1 close parentheses end cell row cell negative 5 end cell 1 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 4 end cell 1 row cell negative 5 end cell 1 end table close parentheses end cell end table 

Mencari invers matriks B:

table attributes columnalign right center left columnspacing 0px end attributes row cell B to the power of negative 1 end exponent end cell equals cell open parentheses table row 7 3 row 2 1 end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell fraction numerator 1 over denominator det open parentheses table row 7 3 row 2 1 end table close parentheses end fraction open parentheses table row 1 cell negative 3 end cell row cell negative 2 end cell 7 end table close parentheses end cell row blank equals cell 1 over 1 open parentheses table row 1 cell negative 3 end cell row cell negative 2 end cell 7 end table close parentheses end cell row blank equals cell open parentheses table row 1 cell negative 3 end cell row cell negative 2 end cell 7 end table close parentheses end cell end table 

Mencari determinan matriks B to the power of negative 1 end exponent A to the power of negative 1 end exponent:

det open parentheses open parentheses table row 1 cell negative 3 end cell row cell negative 2 end cell 7 end table close parentheses open parentheses table row cell negative 4 end cell 1 row cell negative 5 end cell 1 end table close parentheses close parentheses equals det open parentheses table row 11 cell negative 2 end cell row cell negative 27 end cell 5 end table close parentheses equals 11 times 5 minus open parentheses negative 2 close parentheses open parentheses negative 27 close parentheses equals 1  

Jadi, determinan matriks B to the power of negative 1 end exponent A to the power of negative 1 end exponent adalah 1.

Dengan demikian, jawaban yang tepat adalah D.

0

Roboguru

Determinan matriks k yang memenuhi persamaan (43​75​)k=(32​11​) adalah ...

Pembahasan Soal:

Pada persamaan matriks bentuka A X equals B, berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row cell A X end cell equals B row cell A to the power of negative 1 end exponent A X end cell equals cell A to the power of negative 1 end exponent B end cell row cell I X end cell equals cell A to the power of negative 1 end exponent B end cell row X equals cell A to the power of negative 1 end exponent B end cell end table

Invers matriks 2 cross times 2 berlaku:

open parentheses table row a b row c d end table close parentheses to the power of negative 1 end exponent equals fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses

Sehingga diperoleh penyelesaiannya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row 4 7 row 3 5 end table close parentheses k end cell equals cell open parentheses table row 3 1 row 2 1 end table close parentheses end cell row k equals cell open parentheses table row 4 7 row 3 5 end table close parentheses to the power of negative 1 end exponent times open parentheses table row 3 1 row 2 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 4 open parentheses 5 close parentheses minus 7 open parentheses 3 close parentheses end fraction open parentheses table row 5 cell negative 7 end cell row cell negative 3 end cell 4 end table close parentheses open parentheses table row 3 1 row 2 1 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 20 minus 21 end fraction open parentheses table row cell 5 open parentheses 3 close parentheses plus open parentheses negative 7 close parentheses open parentheses 2 close parentheses end cell cell 5 open parentheses 1 close parentheses plus open parentheses negative 7 close parentheses open parentheses 1 close parentheses end cell row cell negative 3 open parentheses 3 close parentheses plus 4 open parentheses 2 close parentheses end cell cell negative 3 open parentheses 1 close parentheses plus 4 open parentheses 1 close parentheses end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 1 end fraction open parentheses table row cell 15 minus 14 end cell cell 5 minus 7 end cell row cell negative 9 plus 8 end cell cell negative 3 plus 4 end cell end table close parentheses end cell row blank equals cell negative 1 times open parentheses table row 1 cell negative 2 end cell row cell negative 1 end cell 1 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell 2 row 1 cell negative 1 end cell end table close parentheses end cell end table

Kemudian tentukan determinan dari matriks k.

Determinan matriks 2 cross times 2 berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row cell open vertical bar table row a b row c d end table close vertical bar end cell equals cell a d minus b c end cell end table

Diperoleh determinan matriks k yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row k equals cell open parentheses table row cell negative 1 end cell 2 row 1 cell negative 1 end cell end table close parentheses end cell row cell open vertical bar k close vertical bar end cell equals cell open vertical bar table row cell negative 1 end cell 2 row 1 cell negative 1 end cell end table close vertical bar end cell row blank equals cell negative 1 open parentheses negative 1 close parentheses minus 2 open parentheses 1 close parentheses end cell row blank equals cell 1 minus 2 end cell row blank equals cell negative 1 end cell end table

Determinan matriks k yang memenuhi persamaan open parentheses table row 4 7 row 3 5 end table close parentheses k equals open parentheses table row 3 1 row 2 1 end table close parentheses adalah negative 1.

Oleh karena itu, jawaban yang benar adalah C.

0

Roboguru

P adalah matriks berordo 2×2 yang memenuhi persamaan: P(2−1​43​)=(58​15−4​) Tentukan ∣P∣.

Pembahasan Soal:

Persamaan matriks bentuk X A equals B berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row cell X A end cell equals B row cell X A A to the power of negative 1 end exponent end cell equals cell B A to the power of negative 1 end exponent end cell row cell X I end cell equals cell B A to the power of negative 1 end exponent end cell row X equals cell B A to the power of negative 1 end exponent end cell end table

Invers matriks 2 cross times 2 berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row a b row c d end table close parentheses end cell row cell A to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator open vertical bar A close vertical bar end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses end cell end table

Determinan matriks 2 cross times 2 berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row a b row c d end table close parentheses end cell row cell open vertical bar A close vertical bar end cell equals cell open vertical bar table row a b row c d end table close vertical bar end cell row blank equals cell a d minus b c end cell end table

Diperoleh penyelesaiannya sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell P open parentheses table row 2 4 row cell negative 1 end cell 3 end table close parentheses end cell equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses end cell row P equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses open parentheses table row 2 4 row cell negative 1 end cell 3 end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses times fraction numerator 1 over denominator 2 open parentheses 3 close parentheses minus 4 open parentheses negative 1 close parentheses end fraction open parentheses table row 3 cell negative 4 end cell row 1 2 end table close parentheses end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses times fraction numerator 1 over denominator 6 plus 4 end fraction open parentheses table row 3 cell negative 4 end cell row 1 2 end table close parentheses end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses times 1 over 10 open parentheses table row 3 cell negative 4 end cell row 1 2 end table close parentheses end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses open parentheses table row cell 1 over 10 times 3 end cell cell 1 over 10 times open parentheses negative 4 close parentheses end cell row cell 1 over 10 times 1 end cell cell 1 over 10 times 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row 5 15 row 8 cell negative 4 end cell end table close parentheses open parentheses table row cell 3 over 10 end cell cell negative 4 over 10 end cell row cell 1 over 10 end cell cell 2 over 10 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 5 open parentheses 3 over 10 close parentheses plus 15 open parentheses 1 over 10 close parentheses end cell cell 5 open parentheses negative 4 over 10 close parentheses plus 15 open parentheses 2 over 10 close parentheses end cell row cell 8 open parentheses 3 over 10 close parentheses plus open parentheses negative 4 close parentheses open parentheses 1 over 10 close parentheses end cell cell 8 open parentheses negative 4 over 10 close parentheses plus open parentheses negative 4 close parentheses open parentheses 2 over 10 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 15 over 10 plus 15 over 10 end cell cell negative 20 over 10 plus 30 over 10 end cell row cell 24 over 10 minus 4 over 10 end cell cell negative 32 over 10 minus 8 over 10 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 30 over 10 end cell cell 10 over 10 end cell row cell 20 over 10 end cell cell negative 40 over 10 end cell end table close parentheses end cell row blank equals cell open parentheses table row 3 1 row 2 cell negative 4 end cell end table close parentheses end cell end table

Kemudian tentukan determinan dari matriks P.

table attributes columnalign right center left columnspacing 0px end attributes row P equals cell open parentheses table row 3 1 row 2 cell negative 4 end cell end table close parentheses end cell row cell open vertical bar P close vertical bar end cell equals cell open vertical bar table row 3 1 row 2 cell negative 4 end cell end table close vertical bar end cell row blank equals cell 3 open parentheses negative 4 close parentheses minus 1 open parentheses 2 close parentheses end cell row blank equals cell negative 12 minus 2 end cell row blank equals cell negative 14 end cell end table

Dengan demikian, open vertical bar P close vertical bar adalah negative 14.

0

Roboguru

Diketahui AX=B, BC=D. Jika , ,  maka X adalah ...

Pembahasan Soal:

Diketahui:

  • A X equals B
  • B C equals D
  • A equals open parentheses table row 1 2 row cell negative 3 end cell cell negative 5 end cell end table close parentheses
  • C equals open parentheses table row 3 2 row 1 1 end table close parentheses
  • D equals open parentheses table row 7 2 row 5 1 end table close parentheses

Ditanya: Maka X adalah ...

Jawab:

Ingat sifat invers pada matriks sebagai berikut

table attributes columnalign right center left columnspacing 0px end attributes row cell X A end cell equals cell B left right double arrow X equals B A to the power of negative 1 end exponent end cell row blank blank blank row cell A X end cell equals cell B left right double arrow X equals A to the power of negative 1 end exponent B end cell end table

Dari sifat invers di atas maka akan ditentukan matriks B dan matriks X. Ingat juga cara menentukan invers dari sebuah matriks yaitu A equals open parentheses table row a b row c d end table close parentheses rightwards arrow A to the power of negative 1 end exponent equals fraction numerator 1 over denominator a d minus b c end fraction open parentheses table row d cell negative b end cell row cell negative c end cell a end table close parentheses.

Akan ditentukan matriks B

table attributes columnalign right center left columnspacing 0px end attributes row cell B C end cell equals D row B equals cell D C to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row 7 2 row 5 1 end table close parentheses open parentheses table row 3 2 row 1 1 end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row 7 2 row 5 1 end table close parentheses times fraction numerator 1 over denominator 3 times 1 minus 2 times 1 end fraction open parentheses table row 1 cell negative 2 end cell row cell negative 1 end cell 3 end table close parentheses end cell row blank equals cell open parentheses table row 7 2 row 5 1 end table close parentheses times fraction numerator 1 over denominator 3 minus 2 end fraction open parentheses table row 1 cell negative 2 end cell row cell negative 1 end cell 3 end table close parentheses end cell row blank equals cell open parentheses table row 7 2 row 5 1 end table close parentheses times 1 over 1 open parentheses table row 1 cell negative 2 end cell row cell negative 1 end cell 3 end table close parentheses end cell row blank equals cell open parentheses table row 7 2 row 5 1 end table close parentheses open parentheses table row 1 cell negative 2 end cell row cell negative 1 end cell 3 end table close parentheses end cell row blank equals cell open parentheses table row cell 7 minus 2 end cell cell negative 14 plus 6 end cell row cell 5 minus 1 end cell cell negative 10 plus 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row 5 cell negative 8 end cell row 4 cell negative 7 end cell end table close parentheses end cell end table

Didapatkan matriks B yaitu open parentheses table row 5 cell negative 8 end cell row 4 cell negative 7 end cell end table close parentheses.

Akan ditentukan matriks X

table attributes columnalign right center left columnspacing 0px end attributes row cell A X end cell equals B row X equals cell A to the power of negative 1 end exponent B end cell row blank equals cell open parentheses table row 1 2 row cell negative 3 end cell cell negative 5 end cell end table close parentheses to the power of negative 1 end exponent open parentheses table row 5 cell negative 8 end cell row 4 cell negative 7 end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 1 times negative 5 minus 2 times negative 3 end fraction open parentheses table row cell negative 5 end cell cell negative 2 end cell row 3 1 end table close parentheses open parentheses table row 5 cell negative 8 end cell row 4 cell negative 7 end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator negative 5 plus 6 end fraction open parentheses table row cell negative 5 end cell cell negative 2 end cell row 3 1 end table close parentheses open parentheses table row 5 cell negative 8 end cell row 4 cell negative 7 end cell end table close parentheses end cell row blank equals cell 1 over 1 open parentheses table row cell negative 5 end cell cell negative 2 end cell row 3 1 end table close parentheses open parentheses table row 5 cell negative 8 end cell row 4 cell negative 7 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 5 end cell cell negative 2 end cell row 3 1 end table close parentheses open parentheses table row 5 cell negative 8 end cell row 4 cell negative 7 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 25 minus 8 end cell cell 40 plus 14 end cell row cell 15 plus 4 end cell cell negative 24 minus 7 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 33 end cell 54 row 19 cell negative 31 end cell end table close parentheses end cell end table

Jadi, dapat disimpulkan bahwa matriks X adalah open parentheses table row cell negative 33 end cell 54 row 19 cell negative 31 end cell end table close parentheses.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

Tapi Roboguru masih mau belajar. Menurut kamu pembahasan kali ini sudah membantu, belum?

Membantu

Kurang Membantu

Apakah pembahasan ini membantu?

Belum menemukan yang kamu cari?

Post pertanyaanmu ke Tanya Jawab, yuk

Mau Bertanya

RUANGGURU HQ

Jl. Dr. Saharjo No.161, Manggarai Selatan, Tebet, Kota Jakarta Selatan, Daerah Khusus Ibukota Jakarta 12860

Coba GRATIS Aplikasi Ruangguru

Produk Ruangguru

Produk Lainnya

Hubungi Kami

Ikuti Kami

©2021 Ruangguru. All Rights Reserved