Pertanyaan

Diketahui matriks B = ( 1 2 2 3 ) dan matriks C = ( 5 8 9 14 ) . Jika C = A B maka A − 1 adalah ...

Diketahui matriks $B = (1223)$ dan matriks $C = (58914)$ . Jika $C = A B$ maka $A^{- 1}$ adalah ...

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Persamaan matriks berlaku: Invers matriks berlaku: Diperoleh penyelesaiannya yaitu: Kemudian tentukan invers dari matriks . Maka adalah . Oleh karena itu, jawaban yang benar adalah D.

Persamaan matriks X A equals B berlaku:

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 .

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

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.