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Pertanyaan

Jika A = ( 2 a 6 ​ 1 1 ​ ) merupakan matriks yang mempunyai invers, maka jumlah semua nilai yang mungkin sehingga det ( − 2 1 ​ A ) = det ( A − 1 ) adalah ...

Jika  merupakan matriks yang mempunyai invers, maka jumlah semua nilai a yang mungkin sehingga  adalah ...

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S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

jawaban yang tepat adalah C.

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Pembahasan

Rumus perkalianskalar dengan matriks yaitu: Rumus invers matriks yaitu: Tentukan terlebih dahulu matriks skalar dari . Kemudian tentukan matriks invers. Kemudian tentukan persamaan dari determinan kedua matriks. Dengan, Jumlah semua nilai yaitu: Maka jumlah semua nilai yang mungkin sehingga adalah . Jadi, jawaban yang tepat adalah C.

Rumus perkalian skalar dengan matriks yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell k times A end cell equals cell k times open parentheses table row a b row c d end table close parentheses end cell row blank equals cell open parentheses table row cell k times a end cell cell k times b end cell row cell k times c end cell cell k times d end cell end table close parentheses end cell end table

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 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

Tentukan terlebih dahulu matriks skalar dari A.

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row cell 2 a end cell 1 row 6 1 end table close parentheses end cell row cell negative 1 half A end cell equals cell open parentheses table row cell negative 1 half open parentheses 2 a close parentheses end cell cell negative 1 half open parentheses 1 close parentheses end cell row cell negative 1 half open parentheses 6 close parentheses end cell cell negative 1 half open parentheses 1 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell negative a end cell cell negative 1 half end cell row cell negative 3 end cell cell negative 1 half end cell end table close parentheses end cell end table

Kemudian tentukan matriks A invers.

table attributes columnalign right center left columnspacing 0px end attributes row A equals cell open parentheses table row cell 2 a end cell 1 row 6 1 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 2 a open parentheses 1 close parentheses minus 1 open parentheses 6 close parentheses end fraction open parentheses table row 1 cell negative 1 end cell row cell negative 6 end cell cell 2 a end cell end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 2 a minus 6 end fraction open parentheses table row 1 cell negative 1 end cell row cell negative 6 end cell cell 2 a end cell end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator 1 over denominator 2 a minus 6 end fraction end cell cell fraction numerator negative 1 over denominator 2 a minus 6 end fraction end cell row cell fraction numerator negative 6 over denominator 2 a minus 6 end fraction end cell cell fraction numerator 2 a over denominator 2 a minus 6 end fraction end cell end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator 1 over denominator 2 a minus 6 end fraction end cell cell fraction numerator negative 1 over denominator 2 a minus 6 end fraction end cell row cell fraction numerator negative 3 over denominator a minus 3 end fraction end cell cell fraction numerator a over denominator a minus 3 end fraction end cell end table close parentheses end cell end table

Kemudian tentukan persamaan dari determinan kedua matriks.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell negative 1 half A end cell equals cell A to the power of negative 1 end exponent end cell row cell open parentheses table row cell negative a end cell cell negative 1 half end cell row cell negative 3 end cell cell negative 1 half end cell end table close parentheses end cell equals cell open parentheses table row cell fraction numerator 1 over denominator 2 a minus 6 end fraction end cell cell fraction numerator negative 1 over denominator 2 a minus 6 end fraction end cell row cell fraction numerator negative 3 over denominator a minus 3 end fraction end cell cell fraction numerator a over denominator a minus 3 end fraction end cell end table close parentheses end cell row cell negative 1 half open parentheses negative a close parentheses minus open parentheses negative 1 half close parentheses open parentheses negative 3 close parentheses end cell equals cell open parentheses fraction numerator 1 over denominator 2 a minus 6 end fraction close parentheses open parentheses fraction numerator a over denominator a minus 3 end fraction close parentheses minus open parentheses fraction numerator negative 1 over denominator 2 a minus 6 end fraction close parentheses open parentheses fraction numerator negative 3 over denominator a minus 3 end fraction close parentheses end cell row cell 1 half a minus 3 over 2 end cell equals cell fraction numerator a over denominator open parentheses 2 a minus 6 close parentheses open parentheses a minus 3 close parentheses end fraction minus fraction numerator 3 over denominator open parentheses 2 a minus 6 close parentheses open parentheses a minus 3 close parentheses end fraction end cell row cell 1 half a minus 3 over 2 end cell equals cell fraction numerator a over denominator 2 open parentheses a minus 3 close parentheses open parentheses a minus 3 close parentheses end fraction minus fraction numerator 3 over denominator 2 open parentheses a minus 3 close parentheses open parentheses a minus 3 close parentheses end fraction end cell row cell 1 half a minus 3 over 2 end cell equals cell fraction numerator a over denominator 2 open parentheses a minus 3 close parentheses squared end fraction minus fraction numerator 3 over denominator 2 open parentheses a minus 3 close parentheses squared end fraction end cell end table end style

table attributes columnalign right center left columnspacing 0px end attributes row cell 1 half a minus 3 over 2 minus fraction numerator a over denominator 2 open parentheses a minus 3 close parentheses squared end fraction plus fraction numerator 3 over denominator 2 open parentheses a minus 3 close parentheses squared end fraction end cell equals 0 row cell fraction numerator a open parentheses a minus 3 close parentheses squared minus 3 open parentheses a minus 3 close parentheses squared minus a plus 3 over denominator 2 open parentheses a minus 3 close parentheses squared end fraction end cell equals 0 row cell fraction numerator a open parentheses a minus 3 close parentheses squared minus 3 open parentheses a minus 3 close parentheses squared minus open parentheses a minus 3 close parentheses over denominator 2 open parentheses a minus 3 close parentheses squared end fraction end cell equals 0 row cell fraction numerator open parentheses a minus 3 close parentheses open parentheses a open parentheses a minus 3 close parentheses minus 3 open parentheses a minus 3 close parentheses minus 1 close parentheses over denominator 2 open parentheses a minus 3 close parentheses squared end fraction end cell equals 0 row cell fraction numerator down diagonal strike open parentheses a minus 3 close parentheses end strike open parentheses a open parentheses a minus 3 close parentheses minus 3 open parentheses a minus 3 close parentheses minus 1 close parentheses over denominator 2 open parentheses a minus 3 close parentheses to the power of down diagonal strike 2 end exponent end fraction end cell equals 0 row cell fraction numerator a open parentheses a minus 3 close parentheses minus 3 open parentheses a minus 3 close parentheses minus 1 over denominator 2 open parentheses a minus 3 close parentheses end fraction end cell equals 0 row cell fraction numerator a squared minus 3 a minus 3 a plus 9 minus 1 over denominator 2 open parentheses a minus 3 close parentheses end fraction end cell equals 0 row cell fraction numerator a squared minus 6 a plus 8 over denominator 2 open parentheses a minus 3 close parentheses end fraction end cell equals 0 row cell a squared minus 6 a plus 8 end cell equals cell 0 open parentheses 2 open parentheses a minus 3 close parentheses close parentheses end cell row cell a squared minus 6 a plus 8 end cell equals 0 row cell a squared minus 4 a minus 2 a plus 8 end cell equals 0 row cell a open parentheses a minus 4 close parentheses minus 2 open parentheses a minus 4 close parentheses end cell equals 0 row cell open parentheses a minus 2 close parentheses open parentheses a minus 4 close parentheses end cell equals 0 end table

Dengan,

table attributes columnalign right center left columnspacing 0px end attributes row cell a subscript 1 minus 2 end cell equals 0 row cell a subscript 1 end cell equals 2 row blank blank blank row cell a subscript 2 minus 4 end cell equals 0 row cell a subscript 2 end cell equals 4 end table

Jumlah semua nilai a yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell a subscript 1 plus a subscript 2 end cell equals cell 2 plus 4 end cell row blank equals 6 end table

Maka jumlah semua nilai a yang mungkin sehingga det space open parentheses negative 1 half A close parentheses equals det space open parentheses A to the power of negative 1 end exponent close parentheses adalah 6.

Jadi, jawaban yang tepat adalah C.

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