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Diketahui begin mathsize 14px style A equals open parentheses table row cell 2 a end cell cell negative 4 end cell row cell negative 1 end cell cell 3 a end cell end table close parentheses end style. Jika matriks A memiliki invers, maka hasil kali semua nilai begin mathsize 14px style a end style yang mungkin sehingga memenuhi begin mathsize 14px style det invisible function application open parentheses A close parentheses equals det invisible function application open parentheses A to the power of negative 1 end exponent close parentheses end style adalah ....

  1. begin mathsize 14px style 5 over 12 end style  

  2. begin mathsize 14px style negative 5 over 3 end style  

  3. begin mathsize 14px style 3 over 5 end style  

  4. begin mathsize 14px style negative 12 over 5 end style  

  5. begin mathsize 14px style 3 end style  

Y. Endah

Master Teacher

Mahasiswa/Alumni Institut Teknologi Bandung

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

Pembahasan

Diketahui undefined,  maka

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell det invisible function application left parenthesis A right parenthesis end cell equals cell 2 a left parenthesis 3 a right parenthesis minus left parenthesis negative 4 right parenthesis left parenthesis negative 1 right parenthesis end cell row blank equals cell 6 a squared minus 4 end cell end table end style   

Karena begin mathsize 14px style det invisible function application left parenthesis A to the power of negative 1 end exponent right parenthesis equals fraction numerator 1 over denominator det invisible function application left parenthesis A right parenthesis end fraction end style, maka

  begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell det invisible function application left parenthesis A right parenthesis end cell equals cell det invisible function application left parenthesis A to the power of negative 1 end exponent right parenthesis end cell row cell det invisible function application left parenthesis A right parenthesis end cell equals cell fraction numerator 1 over denominator det invisible function application left parenthesis A right parenthesis end fraction end cell row cell left parenthesis det invisible function application left parenthesis A right parenthesis right parenthesis squared end cell equals 1 row cell det invisible function application left parenthesis A right parenthesis end cell equals cell plus-or-minus 1 end cell row cell 6 a squared minus 4 end cell equals cell plus-or-minus 1 end cell end table end style    

Pertama, jika begin mathsize 14px style 6 a squared minus 4 equals 1 end style,  maka

begin mathsize 14px style 6 a squared minus 4 equals 1 6 a squared minus 5 equals 0 end style  

Jika persamaan kuadrat begin mathsize 14px style 6 a squared minus 5 equals 0 end style  memiliki akar-akar begin mathsize 14px style a subscript 1 end style dan begin mathsize 14px style a subscript 2 end style, maka

begin mathsize 14px style a subscript 1 a subscript 2 equals negative 5 over 6 end style  

Kedua, jika begin mathsize 14px style 6 a squared minus 4 equals negative 1 end style,  maka

begin mathsize 14px style 6 a squared minus 4 equals negative 1 6 a squared minus 3 equals 0 end style  

Jika persamaan kuadrat begin mathsize 14px style 6 a squared minus 3 equals 0 end style  memiliki akar-akar begin mathsize 14px style a subscript 3 end style dan begin mathsize 14px style a subscript 4 end style, maka

begin mathsize 14px style a subscript 3 a subscript 4 equals negative 3 over 6 equals negative 1 half end style  

Sehingga, hasil kali semua nilai begin mathsize 14px style a end style yang mungkin adalah

begin mathsize 14px style a subscript 1 a subscript 2 a subscript 3 a subscript 4 equals open parentheses negative 5 over 6 close parentheses open parentheses negative 1 half close parentheses equals 5 over 12 end style  

Jadi, jawaban yang tepat adalah A.

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