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Hasil dari  adalah ...

Pertanyaan

Hasil dari open square brackets table row 1 cell negative 7 end cell row 5 9 end table close square brackets cross times open square brackets table row 5 cell negative 3 end cell 8 row 0 2 cell negative 1 end cell end table close square brackets adalah ...

  1. open square brackets table row 5 cell negative 17 end cell 15 row 25 3 31 end table close square brackets

  2. open square brackets table row 25 3 31 row 5 cell negative 17 end cell 15 end table close square brackets

  3. open square brackets table row 5 25 3 row cell negative 17 end cell 31 15 end table close square brackets

  4. open square brackets table row 5 3 15 row 25 cell negative 17 end cell 31 end table close square brackets

  5. open square brackets table row 15 cell negative 17 end cell 5 row 31 3 25 end table close square brackets

Pembahasan Soal:

Perkalian matriks adalah nilai pada matriks yang bisa dihasilkan dengan cara dikalikannya tiap baris dengan setiap kolom yang memiliki jumlah baris yang sama.

Hasil dari open square brackets table row 1 cell negative 7 end cell row 5 9 end table close square brackets cross times open square brackets table row 5 cell negative 3 end cell 8 row 0 2 cell negative 1 end cell end table close square brackets:

open square brackets table row 1 cell negative 7 end cell row 5 9 end table close square brackets cross times open square brackets table row 5 cell negative 3 end cell 8 row 0 2 cell negative 1 end cell end table close square brackets equals open square brackets table row cell 1 open parentheses 5 close parentheses plus open parentheses negative 7 close parentheses 0 end cell cell 1 open parentheses negative 3 close parentheses plus open parentheses negative 7 close parentheses 2 end cell cell 1 open parentheses 8 close parentheses plus open parentheses negative 7 close parentheses open parentheses negative 1 close parentheses end cell row cell 5 open parentheses 5 close parentheses plus 9 open parentheses 0 close parentheses end cell cell 5 open parentheses negative 3 close parentheses plus 9 open parentheses 2 close parentheses end cell cell 5 open parentheses 8 close parentheses plus 9 open parentheses negative 1 close parentheses end cell end table close square brackets equals open square brackets table row cell 5 plus 0 end cell cell negative 3 minus 14 end cell cell 8 plus 7 end cell row cell 25 plus 0 end cell cell negative 15 plus 18 end cell cell 40 minus 9 end cell end table close square brackets equals open square brackets table row 5 cell negative 17 end cell 15 row 25 3 31 end table close square brackets

Sehingga, hasil dari open square brackets table row 1 cell negative 7 end cell row 5 9 end table close square brackets cross times open square brackets table row 5 cell negative 3 end cell 8 row 0 2 cell negative 1 end cell end table close square brackets adalah open square brackets table row 5 cell negative 17 end cell 15 row 25 3 31 end table close square brackets.

Jadi, jawaban yang tepat adalah A.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Dwi

Mahasiswa/Alumni Universitas Sriwijaya

Terakhir diupdate 13 Agustus 2021

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Pertanyaan yang serupa

Jika matriks , maka jumlah semua bilangan yang menjadi unsur pada matriks adalah...

Pembahasan Soal:

Ingatlah operasi perkalian matriks berikut:

A to the power of n equals A times A times.. times A space open parentheses sebanyak space n close parentheses

perhatikan pola berikut:

A squared equals A times A equals open parentheses table row 1 1 0 row 0 1 0 row 0 0 1 end table close parentheses open parentheses table row 1 1 0 row 0 1 0 row 0 0 1 end table close parentheses equals open parentheses table row 1 2 0 row 0 1 0 row 0 0 1 end table close parentheses A cubed equals A squared times A equals open parentheses table row 1 2 0 row 0 1 0 row 0 0 1 end table close parentheses open parentheses table row 1 1 0 row 0 1 0 row 0 0 1 end table close parentheses equals open parentheses table row 1 3 0 row 0 1 0 row 0 0 1 end table close parentheses ...

Berdasarkan hasil tersebut, maka dapat ditentukan

A to the power of 1997 equals open parentheses table row 1 cell 1.997 end cell 0 row 0 1 0 row 0 0 1 end table close parentheses

Jadi, jumlah semua bilangan pada matriks A to the power of 1997 adalah 2.000.

Roboguru

Untuk , , dan  matriks berordo . Buktikan bahwa:

Pembahasan Soal:

table attributes columnalign right center left columnspacing 0px end attributes row cell Misal space straight A end cell equals cell open parentheses table row straight a straight b row straight c straight d end table close parentheses end cell row cell straight A to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator ad minus bc end fraction open parentheses table row straight d cell negative straight b end cell row cell negative straight c end cell straight a end table close parentheses end cell row cell straight A to the power of negative 1 end exponent end cell equals cell open parentheses table row cell fraction numerator straight d over denominator ad minus bc end fraction end cell cell fraction numerator negative straight b over denominator ad minus bc end fraction end cell row cell fraction numerator negative straight c over denominator ad minus bc end fraction end cell cell fraction numerator straight a over denominator ad minus bc end fraction end cell end table close parentheses end cell row cell open parentheses straight A to the power of negative 1 end exponent close parentheses to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator open parentheses fraction numerator straight d over denominator ad minus bc end fraction close parentheses open parentheses fraction numerator straight a over denominator ad minus bc end fraction close parentheses minus open parentheses fraction numerator negative straight c over denominator ad minus bc end fraction close parentheses open parentheses fraction numerator negative straight b over denominator ad minus bc end fraction close parentheses end fraction open parentheses table row cell fraction numerator straight a over denominator ad minus bc end fraction end cell cell fraction numerator straight b over denominator ad minus bc end fraction end cell row cell fraction numerator straight c over denominator ad minus bc end fraction end cell cell fraction numerator straight d over denominator ad minus bc end fraction end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator begin display style ad over open parentheses ad minus bc close parentheses squared end style minus begin display style bc over open parentheses ad minus bc close parentheses squared end style end fraction open parentheses table row cell fraction numerator straight a over denominator ad minus bc end fraction end cell cell fraction numerator straight b over denominator ad minus bc end fraction end cell row cell fraction numerator straight c over denominator ad minus bc end fraction end cell cell fraction numerator straight d over denominator ad minus bc end fraction end cell end table close parentheses end cell row blank equals cell fraction numerator open parentheses ad minus bc close parentheses squared over denominator open parentheses ad minus bc close parentheses end fraction open parentheses table row cell fraction numerator straight a over denominator ad minus bc end fraction end cell cell fraction numerator straight b over denominator ad minus bc end fraction end cell row cell fraction numerator straight c over denominator ad minus bc end fraction end cell cell fraction numerator straight d over denominator ad minus bc end fraction end cell end table close parentheses end cell row blank equals cell fraction numerator open parentheses ad minus bc close parentheses over denominator 1 end fraction open parentheses table row cell fraction numerator straight a over denominator ad minus bc end fraction end cell cell fraction numerator straight b over denominator ad minus bc end fraction end cell row cell fraction numerator straight c over denominator ad minus bc end fraction end cell cell fraction numerator straight d over denominator ad minus bc end fraction end cell end table close parentheses end cell row blank equals cell open parentheses table row straight a straight b row straight c straight d end table close parentheses end cell row cell open parentheses straight A to the power of negative 1 end exponent close parentheses to the power of negative 1 end exponent end cell equals cell straight A space left parenthesis terbukti right parenthesis end cell row blank blank blank end table 

Roboguru

Diketahui matriks  dan matriks . Determinan matriks  adalah ...

Pembahasan Soal:

Diketahui:

A equals open parentheses table row 2 1 row 4 3 end table close parentheses
B equals open parentheses table row cell negative 1 end cell 2 row 0 5 end table close parentheses

Ditanya:

A cross times B

Perkalian matriks adalah nilai pada matriks yang bisa dihasilkan dengan cara dikalikannya tiap baris dengan setiap kolom yang memiliki jumlah baris yang sama. Selain itu, jika terdapat matriks A equals open parentheses table row a b row c d end table close parentheses, maka nilai determinan matriks A dirumuskan det space A equals a d minus b c.

Hasil perkalian dua matriks A dan B:

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

Maka, determinan matriks A cross times B dapat ditentukan seperti berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell det space open parentheses A cross times B close parentheses end cell equals cell det space open parentheses table row cell negative 2 end cell 9 row cell negative 4 end cell 23 end table close parentheses end cell row blank equals cell open parentheses negative 2 close parentheses 23 minus 9 open parentheses negative 4 close parentheses end cell row blank equals cell negative 46 plus 36 end cell row blank equals cell negative 10 end cell end table

Sehingga, determinan matriks A cross times B adalah negative 10.

Jadi, jawaban yang tepat adalah B.

Roboguru

Diberikan . Tentukan matriks  berordo  yang memenuhi persamaan: .

Pembahasan Soal:

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

text det  end text A equals a d minus b c

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

Berdasarkan konsep tersebut dapat ditentukan invers matriks A berikut.

A equals 1 half open parentheses table row 1 1 row 2 cell negative 2 end cell end table close parentheses equals open parentheses table row cell 1 half end cell cell 1 half end cell row 1 cell negative 1 end cell end table close parentheses

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 fraction numerator 1 over denominator negative begin display style 1 half end style minus begin display style 1 half end style end fraction open parentheses table row cell negative 1 end cell cell negative 1 half end cell row cell negative 1 end cell cell 1 half 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 negative 1 end cell cell negative 1 half end cell row cell negative 1 end cell cell 1 half end cell end table close parentheses end cell row blank equals cell open parentheses table row 1 cell 1 half end cell row 1 cell negative 1 half end cell end table close parentheses end cell end table

Misal: Matriks B equals open parentheses table row a b row c d end table close parentheses

Dapat ditentukan perkalian matriks berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell A to the power of negative 1 end exponent times B times A end cell equals cell open parentheses table row 1 0 row 0 3 end table close parentheses end cell row cell open parentheses table row 1 cell 1 half end cell row 1 cell negative 1 half end cell end table close parentheses times open parentheses table row a b row c d end table close parentheses times A end cell equals cell open parentheses table row 1 0 row 0 3 end table close parentheses end cell row cell open parentheses table row cell a plus 1 half c end cell cell b plus 1 half d end cell row cell a minus 1 half c end cell cell b minus 1 half d end cell end table close parentheses end cell equals cell open parentheses table row 1 0 row 0 3 end table close parentheses times A to the power of negative 1 end exponent end cell row cell open parentheses table row cell a plus 1 half c end cell cell b plus 1 half d end cell row cell a minus 1 half c end cell cell b minus 1 half d end cell end table close parentheses end cell equals cell open parentheses table row 1 0 row 0 3 end table close parentheses times open parentheses table row 1 cell 1 half end cell row 1 cell negative 1 half end cell end table close parentheses end cell row cell open parentheses table row cell a plus 1 half c end cell cell b plus 1 half d end cell row cell a minus 1 half c end cell cell b minus 1 half d end cell end table close parentheses end cell equals cell open parentheses table row 1 cell 1 half end cell row 3 cell negative 3 over 2 end cell end table close parentheses end cell end table

Dari kesamaan matriks di atas diperoleh persamaan berikut.

a plus 1 half c equals 1 space left right double arrow space a equals 1 minus 1 half c

a minus 1 half c equals 3

Dengan metode substitusi diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell a minus 1 half c end cell equals 3 row cell open parentheses 1 minus 1 half c close parentheses minus 1 half c end cell equals 3 row cell 1 minus c end cell equals 3 row c equals cell negative 2 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row a equals cell 1 minus 1 half c end cell row blank equals cell 1 minus 1 half open parentheses negative 2 close parentheses end cell row blank equals cell 1 plus 1 end cell row blank equals 2 end table

Dari kesamaan matriks di atas juga diperoleh persamaan berikut.

b plus 1 half d equals 1 half space left right double arrow space b equals 1 half minus 1 half d

b minus 1 half d equals negative 3 over 2

Dengan metode substitusi diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row cell b minus 1 half d end cell equals cell negative 3 over 2 end cell row cell open parentheses 1 half minus 1 half d close parentheses minus 1 half d end cell equals cell negative 3 over 2 end cell row cell 1 half minus d end cell equals cell negative 3 over 2 end cell row d equals cell 1 half plus 3 over 2 end cell row d equals 2 end table

table attributes columnalign right center left columnspacing 0px end attributes row b equals cell 1 half minus 1 half d end cell row blank equals cell 1 half minus 1 half times 2 end cell row blank equals cell 1 half minus 1 end cell row blank equals cell negative 1 half end cell end table

Matriks B equals open parentheses table row a b row c d end table close parentheses equals open parentheses table row 2 cell negative 1 half end cell row cell negative 2 end cell 2 end table close parentheses

Dengan demikian, diperoleh matriks B equals open parentheses table row 2 cell negative 1 half end cell row cell negative 2 end cell 2 end table close parentheses 

Roboguru

maka  ...

Pembahasan Soal:

Perkalian dua buah matiks 2 cross times 2 berlaku:

open parentheses table row a b row c d end table close parentheses open parentheses table row p q row r s end table close parentheses equals open parentheses table row cell a times p plus b times r end cell cell a times q plus b times s end cell row cell c times p plus d times r end cell cell c times q plus d times s end cell end table close parentheses

Sehingga diperoleh matriks A yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row 1 2 row 0 1 end table close parentheses open parentheses table row 2 4 row 1 3 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 open parentheses 2 close parentheses plus 2 open parentheses 1 close parentheses end cell cell 1 open parentheses 4 close parentheses plus 2 open parentheses 3 close parentheses end cell row cell 0 open parentheses 2 close parentheses plus 1 open parentheses 1 close parentheses end cell cell 0 open parentheses 4 close parentheses plus 1 open parentheses 3 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 2 plus 2 end cell cell 4 plus 6 end cell row cell 0 plus 1 end cell cell 0 plus 3 end cell end table close parentheses end cell row blank equals cell open parentheses table row 4 10 row 1 3 end table close parentheses end cell end table

Kemudian tentukan determinan matriks A.

Determinan matriks 2 cross times 2 berlaku:

open vertical bar table row a b row c d end table close vertical bar equals a d minus b c

Sehingga diperoleh determinan matriks A yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row 4 10 row 1 3 end table close parentheses end cell row cell open vertical bar A close vertical bar end cell equals cell open vertical bar table row 4 10 row 1 3 end table close vertical bar end cell row blank equals cell 4 open parentheses 3 close parentheses minus 10 open parentheses 1 close parentheses end cell row blank equals cell 12 minus 10 end cell row blank equals 2 end table

Maka open vertical bar A close vertical bar equals2.

Oleh karena itu, jawaban yang benar adalah A.

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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