Pertanyaan

Jika A = ( 1 a a 2 ) merupakan matriks yang mempunyai invers, maka hasil kali semua nilai yang mungkin sehingga det ( A − 1 ) = det ( A 3 ) adalah ...

Jika $A = (1 a a 2)$ merupakan matriks yang mempunyai invers, maka hasil kali semua nilai $a$ yang mungkin sehingga $det (A^{- 1}) = det (A^{3})$ adalah ...

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

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Jawaban

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Pembahasan

Rumus invers matriks yaitu: Rumus determinan matriks yaitu: Tentukan terlebih dahulu invers matriks , yaitu: Kemudian tentukan nilai determinannya. Kemudian tentukan hasil perkalian pangkat dari matriks . Kemudian tentukan nilai determinannya. Sehingga diperoleh perhitungannya sebagai berikut: Nilai dan Hasil kali semua nilai : Maka hasil kali semua nilai yang mungkin sehingga adalah . Jadi, jawaban tidak ada yang tepat

Rumus invers matriks A equals open parentheses table row a b row c d end table close parentheses yaitu:

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

Rumus determinan 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 det space A end cell equals cell a times d minus b times c end cell end table

Tentukan terlebih dahulu invers matriks , yaitu:

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

Kemudian tentukan nilai determinannya.

$table attributes columnalign right center left columnspacing 0px end attributes row cell A to the power of negative 1 end exponent end cell equals cell open parentheses table row cell fraction numerator 2 over denominator 2 minus a squared end fraction end cell cell negative fraction numerator a over denominator 2 minus a squared end fraction end cell row cell negative fraction numerator a over denominator 2 minus a squared end fraction end cell cell fraction numerator 1 over denominator 2 minus a squared end fraction end cell end table close parentheses end cell row cell det space A to the power of negative 1 end exponent end cell equals cell open parentheses fraction numerator 2 over denominator 2 minus a squared end fraction close parentheses open parentheses fraction numerator 1 over denominator 2 minus a squared end fraction close parentheses minus open parentheses negative fraction numerator a over denominator 2 minus a squared end fraction close parentheses open parentheses negative fraction numerator a over denominator 2 minus a squared end fraction close parentheses end cell row blank equals cell fraction numerator down diagonal strike 2 minus a squared end strike over denominator open parentheses 2 minus a squared close parentheses to the power of down diagonal strike 2 end exponent end fraction end cell row blank equals cell fraction numerator 1 over denominator 2 minus a squared end fraction end cell end table$

Kemudian tentukan hasil perkalian pangkat dari matriks .

Kemudian tentukan nilai determinannya.

Sehingga diperoleh perhitungannya sebagai berikut:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell det space open parentheses A to the power of negative 1 end exponent close parentheses end cell equals cell det space open parentheses A close parentheses cubed end cell row cell fraction numerator 1 over denominator 2 minus a squared end fraction end cell equals cell negative a to the power of 6 plus 6 a to the power of 4 minus 12 a squared plus 8 end cell row cell fraction numerator 1 over denominator 2 minus a squared end fraction minus open parentheses negative a to the power of 6 plus 6 a to the power of 4 minus 12 a squared plus 8 close parentheses end cell equals 0 row cell fraction numerator 1 plus a to the power of 6 open parentheses 2 minus a squared close parentheses minus 6 a to the power of 4 open parentheses 2 minus a squared close parentheses plus 12 a squared open parentheses 2 minus a squared close parentheses minus 8 open parentheses 2 minus a squared close parentheses over denominator 2 minus a squared end fraction end cell equals 0 row cell fraction numerator 1 plus 2 a to the power of 6 minus a to the power of 8 minus 12 a to the power of 4 plus 6 a to the power of 6 plus 24 a squared minus 12 a to the power of 4 minus 12 plus 8 a squared over denominator 2 minus a squared end fraction end cell equals 0 row cell fraction numerator negative a to the power of 8 plus 2 a to the power of 6 plus 6 a to the power of 6 minus 12 a to the power of 4 minus 12 a to the power of 4 plus 24 a squared plus 8 a squared plus 1 minus 12 over denominator 2 minus a squared end fraction end cell equals 0 row cell fraction numerator negative a to the power of 8 plus 8 a to the power of 6 minus 24 a to the power of 4 plus 32 a squared minus 11 over denominator 2 minus a squared end fraction end cell equals 0 row cell 2 minus a squared end cell equals 0 row cell negative a squared end cell equals cell negative 2 end cell row cell a squared end cell equals 2 row a equals cell plus-or-minus square root of 2 end cell end table end style$

Nilai a equals square root of 2 dan a equals negative square root of 2

Hasil kali semua nilai :

table attributes columnalign right center left columnspacing 0px end attributes row cell a subscript 1 times a subscript 2 end cell equals cell square root of 2 times negative square root of 2 end cell row blank equals cell negative 2 end cell end table

Maka hasil kali semua nilai yang mungkin sehingga det space open parentheses A to the power of negative 1 end exponent close parentheses equals det space open parentheses A cubed close parentheses adalah negative 2 .

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