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Diketahui matriks , , jika  maka nilai ...

Pertanyaan

Diketahui matriks begin mathsize 14px style straight A equals open parentheses table row 3 2 row 6 x end table close parentheses end stylebegin mathsize 14px style straight B equals open parentheses table row cell x over 3 end cell cell negative 2 end cell row cell negative 2 over 3 end cell 1 end table close parentheses end style, jika begin mathsize 14px style straight A to the power of negative 1 end exponent equals straight B to the power of straight t end style maka nilai begin mathsize 14px style x equals end style...

Pembahasan Soal:

Gunakan konsep invers matriks, begin mathsize 14px style straight A equals open parentheses table row 3 2 row 6 x end table close parentheses end style maka invers matriks A adalah

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight A to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator det space straight A end fraction times adj space straight A end cell row blank equals cell fraction numerator 1 over denominator 3 times x minus 2 times 6 end fraction times open parentheses table row x cell negative 2 end cell row cell negative 6 end cell 3 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 3 x minus 12 end fraction times open parentheses table row x cell negative 2 end cell row cell negative 6 end cell 3 end table close parentheses end cell row blank equals cell open parentheses table row cell fraction numerator x over denominator 3 x minus 12 end fraction end cell cell fraction numerator negative 2 over denominator 3 x minus 12 end fraction end cell row cell fraction numerator negative 6 over denominator 3 x minus 12 end fraction end cell cell fraction numerator 3 over denominator 3 x minus 12 end fraction end cell end table close parentheses end cell end table end style 

Jika begin mathsize 14px style straight A to the power of negative 1 end exponent equals straight B to the power of straight t end style, gunakan konsep transpose matriks sehingga:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight A to the power of negative 1 end exponent end cell equals cell straight B to the power of straight t end cell row cell open parentheses table row cell fraction numerator x over denominator 3 x minus 12 end fraction end cell cell fraction numerator negative 2 over denominator 3 x minus 12 end fraction end cell row cell fraction numerator negative 6 over denominator 3 x minus 12 end fraction end cell cell fraction numerator 3 over denominator 3 x minus 12 end fraction end cell end table close parentheses end cell equals cell open parentheses table row cell x over 3 end cell cell negative 2 over 3 end cell row cell negative 2 end cell 1 end table close parentheses end cell end table end style 

Berdasarkan konsep kesamaan matriks maka diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator negative 6 over denominator 3 x minus 12 end fraction end cell equals cell negative 2 end cell row cell negative 6 end cell equals cell negative 2 open parentheses 3 x minus 12 close parentheses end cell row cell negative 6 end cell equals cell negative 6 x plus 24 end cell row cell 6 x end cell equals cell 24 plus 6 end cell row cell 6 x end cell equals 30 row x equals cell 30 over 6 end cell row x equals 5 end table end style  

Dengan demikian, nilai dari begin mathsize 14px style x end style adalah 5.

Pembahasan terverifikasi oleh Roboguru

Terakhir diupdate 12 April 2021

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