Roboguru

Jika A matriks berordo 2×2 sehingga A(1−1​)=(−15​) dan A(21​)=(47​), maka A2= ...

Pertanyaan

Jika A matriks berordo 2 cross times 2 sehingga A open parentheses table row 1 row cell negative 1 end cell end table close parentheses equals open parentheses table row cell negative 1 end cell row 5 end table close parentheses dan A open parentheses table row 2 row 1 end table close parentheses equals open parentheses table row 4 row 7 end table close parentheses, maka A squared equals ...

  1. open parentheses table row 1 2 row 4 cell negative 1 end cell end table close parentheses

  2. open parentheses table row 9 0 row 0 9 end table close parentheses

  3. open parentheses table row 7 0 row 0 9 end table close parentheses

  4. open parentheses table row 7 0 row 0 9 end table close parentheses

  5. open parentheses table row 7 0 row 0 7 end table close parentheses

Pembahasan Soal:

Misal: P adalah matriks table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses table row 1 2 row cell negative 1 end cell 1 end table close parentheses end cell end table dan Q adalah matriks open parentheses table row cell negative 1 end cell 4 row 5 7 end table close parentheses.

Sehingga pada soal terbentuk operasi matriks A P equals Q.

Dengan menggunakan sifat invers matriks, diperoleh:

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

Sehingga diperoleh penyelesaiannya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell Q P to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row cell negative 1 end cell 4 row 5 7 end table close parentheses open parentheses table row 1 2 row cell negative 1 end cell 1 end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row cell negative 1 end cell 4 row 5 7 end table close parentheses times fraction numerator 1 over denominator 1 open parentheses 1 close parentheses minus 2 open parentheses negative 1 close parentheses end fraction open parentheses table row 1 cell negative 2 end cell row 1 1 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell 4 row 5 7 end table close parentheses times fraction numerator 1 over denominator 1 plus 2 end fraction open parentheses table row 1 cell negative 2 end cell row 1 1 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell 4 row 5 7 end table close parentheses times 1 third open parentheses table row 1 cell negative 2 end cell row 1 1 end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell 4 row 5 7 end table close parentheses times open parentheses table row cell 1 third open parentheses 1 close parentheses end cell cell 1 third open parentheses negative 2 close parentheses end cell row cell 1 third open parentheses 1 close parentheses end cell cell 1 third open parentheses 1 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 end cell 4 row 5 7 end table close parentheses times open parentheses table row cell 1 third end cell cell negative 2 over 3 end cell row cell 1 third end cell cell 1 third end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 open parentheses 1 third close parentheses plus 4 open parentheses 1 third close parentheses end cell cell negative 1 open parentheses negative 2 over 3 close parentheses plus 4 open parentheses 1 third close parentheses end cell row cell 5 open parentheses 1 third close parentheses plus 7 open parentheses 1 third close parentheses end cell cell 5 open parentheses negative 2 over 3 close parentheses plus 7 open parentheses 1 third close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 1 third plus 4 over 3 end cell cell 2 over 3 plus 4 over 3 end cell row cell 5 over 3 plus 7 over 3 end cell cell negative 10 over 3 plus 7 over 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 3 over 3 end cell cell 6 over 3 end cell row cell 12 over 3 end cell cell negative 3 over 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row 1 2 row 4 cell negative 1 end cell end table close parentheses end cell end table

Diperoleh matriks A equals open parentheses table row 1 2 row 4 cell negative 1 end cell end table close parentheses.

Maka,

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row 1 2 row 4 cell negative 1 end cell end table close parentheses end cell row cell A squared end cell equals cell open parentheses table row 1 2 row 4 cell negative 1 end cell end table close parentheses open parentheses table row 1 2 row 4 cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 1 open parentheses 1 close parentheses plus 2 open parentheses 4 close parentheses end cell cell 1 open parentheses 2 close parentheses plus 2 open parentheses negative 1 close parentheses end cell row cell 4 open parentheses 1 close parentheses plus open parentheses negative 1 close parentheses open parentheses 4 close parentheses end cell cell 4 open parentheses 2 close parentheses plus open parentheses negative 1 close parentheses open parentheses negative 1 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 1 plus 8 end cell cell 2 minus 2 end cell row cell 4 minus 4 end cell cell 8 plus 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row 9 0 row 0 9 end table close parentheses end cell end table

Maka A squared equalsopen parentheses table row 9 0 row 0 9 end table close parentheses.

Oleh karena itu, jawaban yang benar adalah B.

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

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Pertanyaan yang serupa

Jika X(25​41​)=(1231​617​) maka X= ...

Pembahasan Soal:

Pada 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:

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 X open parentheses table row 2 4 row 5 1 end table close parentheses end cell equals cell open parentheses table row 12 6 row 31 17 end table close parentheses end cell row X equals cell open parentheses table row 12 6 row 31 17 end table close parentheses open parentheses table row 2 4 row 5 1 end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row 12 6 row 31 17 end table close parentheses times fraction numerator 1 over denominator 2 open parentheses 1 close parentheses minus 4 open parentheses 5 close parentheses end fraction open parentheses table row 1 cell negative 4 end cell row cell negative 5 end cell 2 end table close parentheses end cell row blank equals cell open parentheses table row 12 6 row 31 17 end table close parentheses times fraction numerator 1 over denominator 2 minus 20 end fraction open parentheses table row 1 cell negative 4 end cell row cell negative 5 end cell 2 end table close parentheses end cell row blank equals cell open parentheses table row 12 6 row 31 17 end table close parentheses times fraction numerator 1 over denominator negative 18 end fraction open parentheses table row 1 cell negative 4 end cell row cell negative 5 end cell 2 end table close parentheses end cell row blank equals cell open parentheses table row 12 6 row 31 17 end table close parentheses open parentheses table row cell negative 1 over 18 end cell cell fraction numerator negative 4 over denominator negative 18 end fraction end cell row cell fraction numerator negative 5 over denominator negative 18 end fraction end cell cell fraction numerator 2 over denominator negative 18 end fraction end cell end table close parentheses end cell row blank equals cell open parentheses table row 12 6 row 31 17 end table close parentheses open parentheses table row cell negative 1 over 18 end cell cell 2 over 9 end cell row cell 5 over 18 end cell cell negative 1 over 9 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 12 open parentheses negative 1 over 18 close parentheses plus 6 open parentheses 5 over 18 close parentheses end cell cell 12 open parentheses 2 over 9 close parentheses plus 6 open parentheses negative 1 over 9 close parentheses end cell row cell 31 open parentheses negative 1 over 18 close parentheses plus 17 open parentheses 5 over 18 close parentheses end cell cell 31 open parentheses 2 over 9 close parentheses plus 17 open parentheses negative 1 over 9 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 12 over 18 plus 30 over 18 end cell cell 24 over 9 minus 6 over 9 end cell row cell negative 31 over 18 plus 85 over 18 end cell cell 62 over 9 minus 17 over 9 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 18 over 18 end cell cell 18 over 9 end cell row cell 54 over 18 end cell cell 45 over 9 end cell end table close parentheses end cell row blank equals cell open parentheses table row 1 2 row 3 5 end table close parentheses end cell end table

Maka X equalsopen parentheses table row 1 2 row 3 5 end table close parentheses.

Oleh karena itu, jawaban yang benar adalah B.

0

Roboguru

Jika matriks A=(−51​73​), matriks C=(13−26​−2346​), dan BA=C, matriks B adalah ....

Pembahasan Soal:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell B A end cell equals C row cell B A A to the power of negative 1 end exponent end cell equals cell C A to the power of negative 1 end exponent end cell row B equals cell C A to the power of negative 1 end exponent end cell row B equals cell open parentheses table row 13 cell negative 23 end cell row cell negative 26 end cell 46 end table close parentheses open parentheses table row cell negative 5 end cell 7 row 1 3 end table close parentheses to the power of negative 1 end exponent end cell row B equals cell open parentheses table row 13 cell negative 23 end cell row cell negative 26 end cell 46 end table close parentheses fraction numerator 1 over denominator 15 minus 7 end fraction open parentheses table row cell negative 3 end cell cell negative 7 end cell row cell negative 1 end cell cell negative 5 end cell end table close parentheses end cell row B equals cell 1 over 8 open parentheses table row 13 cell negative 23 end cell row cell negative 26 end cell 46 end table close parentheses open parentheses table row cell negative 3 end cell cell negative 7 end cell row cell negative 1 end cell cell negative 5 end cell end table close parentheses end cell row B equals cell 1 over 8 open parentheses table row cell negative 39 plus 23 end cell cell negative 91 plus 115 end cell row cell 78 minus 46 end cell cell 182 minus 230 end cell end table close parentheses end cell row B equals cell 1 over 8 open parentheses table row cell negative 16 end cell 24 row 32 cell negative 48 end cell end table close parentheses end cell row B equals cell open parentheses table row cell negative 2 end cell 3 row 4 cell negative 6 end cell end table close parentheses end cell end table end style 

Jadi, jawaban yang tepat adalah D.

1

Roboguru

Diketahui matriks B=(12​23​) dan matriks C=(58​914​). Jika C=AB maka A−1 adalah ...

Pembahasan Soal:

Persamaan matriks 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

Diperoleh penyelesaiannya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row C equals cell A B end cell row cell C B to the power of negative 1 end exponent end cell equals cell A B B to the power of negative 1 end exponent end cell row cell C B to the power of negative 1 end exponent end cell equals cell A I end cell row cell C B to the power of negative 1 end exponent end cell equals A row cell open parentheses table row 5 9 row 8 14 end table close parentheses open parentheses table row 1 2 row 2 3 end table close parentheses to the power of negative 1 end exponent end cell equals A row cell open parentheses table row 5 9 row 8 14 end table close parentheses times fraction numerator 1 over denominator 1 open parentheses 3 close parentheses minus 2 open parentheses 2 close parentheses end fraction open parentheses table row 3 cell negative 2 end cell row cell negative 2 end cell 1 end table close parentheses end cell equals A row cell open parentheses table row 5 9 row 8 14 end table close parentheses times fraction numerator 1 over denominator 3 minus 4 end fraction open parentheses table row 3 cell negative 2 end cell row cell negative 2 end cell 1 end table close parentheses end cell equals A row cell open parentheses table row 5 9 row 8 14 end table close parentheses times fraction numerator 1 over denominator negative 1 end fraction open parentheses table row 3 cell negative 2 end cell row cell negative 2 end cell 1 end table close parentheses end cell equals A row cell open parentheses table row 5 9 row 8 14 end table close parentheses times negative 1 times open parentheses table row 3 cell negative 2 end cell row cell negative 2 end cell 1 end table close parentheses end cell equals A row cell open parentheses table row 5 9 row 8 14 end table close parentheses open parentheses table row cell negative 3 end cell 2 row 2 cell negative 1 end cell end table close parentheses end cell equals A row cell open parentheses table row cell 5 open parentheses negative 3 close parentheses plus 9 open parentheses 2 close parentheses end cell cell 5 open parentheses 2 close parentheses plus 9 open parentheses negative 1 close parentheses end cell row cell 8 open parentheses negative 3 close parentheses plus 14 open parentheses 2 close parentheses end cell cell 8 open parentheses 2 close parentheses plus 14 open parentheses negative 1 close parentheses end cell end table close parentheses end cell equals A row cell open parentheses table row cell negative 15 plus 18 end cell cell 10 minus 9 end cell row cell negative 24 plus 28 end cell cell 16 minus 14 end cell end table close parentheses end cell equals A row cell open parentheses table row 3 1 row 4 2 end table close parentheses end cell equals A end table

Kemudian tentukan invers dari matriks A.

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row 3 1 row 4 2 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 3 open parentheses 2 close parentheses minus 1 open parentheses 4 close parentheses end fraction open parentheses table row 2 cell negative 1 end cell row cell negative 4 end cell 3 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 6 minus 4 end fraction open parentheses table row 2 cell negative 1 end cell row cell negative 4 end cell 3 end table close parentheses end cell row blank equals cell 1 half open parentheses table row 2 cell negative 1 end cell row cell negative 4 end cell 3 end table close parentheses end cell row blank equals cell open parentheses table row cell 2 over 2 end cell cell fraction numerator negative 1 over denominator 2 end fraction end cell row cell fraction numerator negative 4 over denominator 2 end fraction end cell cell 3 over 2 end cell end table close parentheses end cell row blank equals cell open parentheses table row 1 cell negative 1 half end cell row cell negative 2 end cell cell 3 over 2 end cell end table close parentheses end cell end table

Maka A to the power of negative 1 end exponent adalah open parentheses table row 1 cell negative 1 half end cell row cell negative 2 end cell cell 3 over 2 end cell end table close parentheses.

Oleh karena itu, jawaban yang benar adalah D.

0

Roboguru

Diketahui matriks A=(−44​−2p​), B=(−13​8−4​), dan C=(−214​−248​). Jika AB=C, nilai 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

Sehingga diperoleh penyelesaiannya yaitu:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell A B end cell equals C row cell open parentheses table row cell negative 4 end cell cell negative 2 end cell row 4 p end table close parentheses open parentheses table row cell negative 1 end cell 8 row 3 cell negative 4 end cell end table close parentheses end cell equals cell open parentheses table row cell negative 2 end cell cell negative 24 end cell row 14 8 end table close parentheses end cell row cell open parentheses table row cell negative 4 end cell cell negative 2 end cell row 4 p end table close parentheses end cell equals cell open parentheses table row cell negative 2 end cell cell negative 24 end cell row 14 8 end table close parentheses open parentheses table row cell negative 1 end cell 8 row 3 cell negative 4 end cell end table close parentheses to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row cell negative 2 end cell cell negative 24 end cell row 14 8 end table close parentheses times fraction numerator 1 over denominator negative 1 open parentheses negative 4 close parentheses minus 8 open parentheses 3 close parentheses end fraction open parentheses table row cell negative 4 end cell cell negative 8 end cell row cell negative 3 end cell cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 end cell cell negative 24 end cell row 14 8 end table close parentheses times fraction numerator 1 over denominator 4 minus 24 end fraction open parentheses table row cell negative 4 end cell cell negative 8 end cell row cell negative 3 end cell cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 end cell cell negative 24 end cell row 14 8 end table close parentheses times open parentheses negative 1 over 20 close parentheses open parentheses table row cell negative 4 end cell cell negative 8 end cell row cell negative 3 end cell cell negative 1 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 end cell cell negative 24 end cell row 14 8 end table close parentheses open parentheses table row cell fraction numerator negative 4 over denominator negative 20 end fraction end cell cell fraction numerator negative 8 over denominator negative 20 end fraction end cell row cell fraction numerator negative 3 over denominator negative 20 end fraction end cell cell fraction numerator negative 1 over denominator negative 20 end fraction end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 end cell cell negative 24 end cell row 14 8 end table close parentheses open parentheses table row cell 1 fifth end cell cell 2 over 5 end cell row cell 3 over 20 end cell cell 1 over 20 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 open parentheses 1 fifth close parentheses plus open parentheses negative 24 close parentheses open parentheses 3 over 20 close parentheses end cell cell negative 2 open parentheses 2 over 5 close parentheses plus open parentheses negative 24 close parentheses open parentheses 1 over 20 close parentheses end cell row cell 14 open parentheses 1 fifth close parentheses plus 8 open parentheses 3 over 20 close parentheses end cell cell 14 open parentheses 2 over 5 close parentheses plus 8 open parentheses 1 over 20 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 2 over 5 minus 18 over 5 end cell cell negative 4 over 5 minus 6 over 5 end cell row cell 14 over 5 plus 6 over 5 end cell cell 28 over 5 plus 2 over 5 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 20 over 5 end cell cell negative 10 over 5 end cell row cell 20 over 5 end cell cell 30 over 5 end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative 4 end cell cell negative 2 end cell row 4 6 end table close parentheses end cell end table end style

Nilai p equals6.

Oleh karena itu, jawaban yang benar adalah E.

0

Roboguru

Jika B=(13​25​) dan AB−1=(24​13​), maka A= ...

Pembahasan Soal:

Pada persamaan matriks 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

Sehingga diperoleh penyelesaiannya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell A B to the power of negative 1 end exponent end cell equals cell open parentheses table row 2 1 row 4 3 end table close parentheses end cell row A equals cell open parentheses table row 2 1 row 4 3 end table close parentheses times open parentheses B to the power of negative 1 end exponent close parentheses to the power of negative 1 end exponent end cell row blank equals cell open parentheses table row 2 1 row 4 3 end table close parentheses times B to the power of 1 end cell row blank equals cell open parentheses table row 2 1 row 4 3 end table close parentheses times B end cell row blank equals cell open parentheses table row 2 1 row 4 3 end table close parentheses open parentheses table row 1 2 row 3 5 end table close parentheses end cell row blank equals cell open parentheses table row cell 2 open parentheses 1 close parentheses plus 1 open parentheses 3 close parentheses end cell cell 2 open parentheses 2 close parentheses plus 1 open parentheses 5 close parentheses end cell row cell 4 open parentheses 1 close parentheses plus 3 open parentheses 3 close parentheses end cell cell 4 open parentheses 2 close parentheses plus 3 open parentheses 5 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 2 plus 3 end cell cell 4 plus 5 end cell row cell 4 plus 9 end cell cell 8 plus 15 end cell end table close parentheses end cell row blank equals cell open parentheses table row 5 9 row 13 23 end table close parentheses end cell end table

Maka A equalsopen parentheses table row 5 9 row 13 23 end table close parentheses.

Oleh karena itu, jawaban yang benar adalah A.

0

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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

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