Diketahui A=(3172), B=(41), dan C=(xy). Jika A−1BT=C, nilai x−y= ...

Pertanyaan

Diketahui $A=\begin{pmatrix}3&7\\1&2\end{pmatrix}$ , $B=\begin{pmatrix}4&1\end{pmatrix}$ , dan $C=\begin{pmatrix}x\\y\end{pmatrix}$ . Jika $A^{-1}B^T=C$ , nilai $x-y=$ ...

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Rumus invers matriks 2 cross times 2 yaitu:

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

Rumus transpose matriks 2 cross times 1 yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row a b end table close parentheses end cell row cell A to the power of T end cell equals cell open parentheses table row a row b end table close parentheses end cell end table

Tentukan terlebih dahulu transpose dari matriks .

table attributes columnalign right center left columnspacing 0px end attributes row B equals cell open parentheses table row 4 1 end table close parentheses end cell row cell B to the power of T end cell equals cell open parentheses table row 4 row 1 end table close parentheses end cell end table

Diperoleh penyelesainnya yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell A to the power of negative 1 end exponent B to the power of T end cell equals C row cell open parentheses table row 3 7 row 1 2 end table close parentheses to the power of negative 1 end exponent times open parentheses table row 4 row 1 end table close parentheses end cell equals cell open parentheses table row x row y end table close parentheses end cell row cell fraction numerator 1 over denominator 3 open parentheses 2 close parentheses minus 7 open parentheses 1 close parentheses end fraction open parentheses table row 2 cell negative 7 end cell row cell negative 1 end cell 3 end table close parentheses times open parentheses table row 4 row 1 end table close parentheses end cell equals cell open parentheses table row x row y end table close parentheses end cell row cell fraction numerator 1 over denominator 6 minus 7 end fraction times open parentheses table row cell 2 open parentheses 4 close parentheses plus open parentheses negative 7 close parentheses open parentheses 1 close parentheses end cell row cell negative 1 open parentheses 4 close parentheses plus 3 open parentheses 1 close parentheses end cell end table close parentheses end cell equals cell open parentheses table row x row y end table close parentheses end cell row cell negative 1 times open parentheses table row cell 8 minus 7 end cell row cell negative 4 plus 3 end cell end table close parentheses end cell equals cell open parentheses table row x row y end table close parentheses end cell row cell negative 1 times open parentheses table row 1 row cell negative 1 end cell end table close parentheses end cell equals cell open parentheses table row x row y end table close parentheses end cell row cell open parentheses table row cell negative 1 end cell row 1 end table close parentheses end cell equals cell open parentheses table row x row y end table close parentheses end cell end table$

Pada perhitungan matriks, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell negative 1 end cell end table dan

Dipereoleh nilai:

table attributes columnalign right center left columnspacing 0px end attributes row cell x minus y end cell equals cell negative 1 minus 1 end cell row blank equals cell negative 2 end cell end table

Nilai x minus y equals negative 2 .

Oleh karena itu, jawaban yang benar adalah D.