Pertanyaan

Diketahui matriks A = ( 4 3 − 9 − 4 p ) , B = ( 5 p 1 − 5 3 ) , dan C = ( − 10 − 4 8 6 p ) . Jika matriks A − B = C − 1 , maka nilai 4 p = ...

Diketahui matriks $A = (43 - 9 - 4 p)$ , $B = (5 p 1 - 5 3)$ , dan $C = (- 10 - 4 8 6 p)$ . Jika matriks $A - B = C^{- 1}$ , maka nilai $4 p =$ ...

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah A.

Pembahasan

Rumus selisih dua buah matriks yaitu: Rumus invers matriks yaitu: Diperoleh penyelesaiannya yaitu: Perhatikan pada perhitungan matriks terdapat persamaan-persamaan. Misal diambil persamaan , maka nilai yaitu: Sehingga diperoleh: Maka nilai . Oleh karena itu, jawaban yang benar adalah A.

Rumus selisih dua buah matriks 2 cross times 2 yaitu:

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

Rumus invers matriks 2 cross times 2 yaitu:

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

Diperoleh penyelesaiannya yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row cell A minus B end cell equals cell C to the power of negative 1 end exponent end cell row cell open parentheses table row 4 cell negative 9 end cell row 3 cell negative 4 p end cell end table close parentheses minus open parentheses table row cell 5 p end cell cell negative 5 end cell row 1 3 end table close parentheses end cell equals cell open parentheses table row cell negative 10 end cell 8 row cell negative 4 end cell cell 6 p end cell end table close parentheses to the power of negative 1 end exponent end cell row cell open parentheses table row cell 4 minus 5 p end cell cell negative 9 minus open parentheses negative 5 close parentheses end cell row cell 3 minus 1 end cell cell negative 4 p minus 3 end cell end table close parentheses end cell equals cell fraction numerator 1 over denominator negative 10 open parentheses 6 p close parentheses minus 8 open parentheses negative 4 close parentheses end fraction open parentheses table row cell 6 p end cell cell negative 8 end cell row 4 cell negative 10 end cell end table close parentheses end cell row cell open parentheses table row cell 4 minus 5 p end cell cell negative 9 plus 5 end cell row 2 cell negative 4 p minus 3 end cell end table close parentheses end cell equals cell fraction numerator 1 over denominator negative 60 p plus 32 end fraction open parentheses table row cell 6 p end cell cell negative 8 end cell row 4 cell negative 10 end cell end table close parentheses end cell row cell open parentheses table row cell 4 minus 5 p end cell cell negative 4 end cell row 2 cell negative 4 p minus 3 end cell end table close parentheses end cell equals cell open parentheses table row cell fraction numerator 6 p over denominator negative 60 p plus 32 end fraction end cell cell fraction numerator negative 8 over denominator negative 60 p plus 32 end fraction end cell row cell fraction numerator 4 over denominator negative 60 p plus 32 end fraction end cell cell fraction numerator negative 10 over denominator negative 60 p plus 32 end fraction end cell end table close parentheses end cell row cell open parentheses table row cell 4 minus 5 p end cell cell negative 4 end cell row 2 cell negative 4 p minus 3 end cell end table close parentheses end cell equals cell open parentheses table row cell fraction numerator 3 p over denominator negative 30 p plus 16 end fraction end cell cell fraction numerator negative 2 over denominator negative 15 p plus 8 end fraction end cell row cell fraction numerator 1 over denominator negative 15 p plus 8 end fraction end cell cell fraction numerator negative 5 over denominator negative 30 p plus 16 end fraction end cell end table close parentheses end cell end table$

Perhatikan pada perhitungan matriks terdapat persamaan-persamaan.

Misal diambil persamaan $2 equals fraction numerator 1 over denominator negative 15 p plus 8 end fraction$ , maka nilai yaitu:

$table attributes columnalign right center left columnspacing 0px end attributes row 2 equals cell fraction numerator 1 over denominator negative 15 p plus 8 end fraction end cell row cell 2 open parentheses negative 15 p plus 8 close parentheses end cell equals 1 row cell negative 30 p plus 16 end cell equals 1 row cell negative 30 p end cell equals cell 1 minus 16 end cell row cell negative 30 p end cell equals cell negative 15 end cell row p equals cell fraction numerator negative 15 over denominator negative 30 end fraction end cell row p equals cell 1 half end cell end table$

Sehingga diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 p end cell equals cell 4 times 1 half end cell row blank equals 2 end table

Maka nilai 4 p equals .

Oleh karena itu, jawaban yang benar adalah A.