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Selesaikan sistem persamaan berikut dengan menggunakan matriks. ⎩ ⎨ ⎧ ​ x + 2 y − z = 8 2 x − y = 0 2 x − y + 3 z = − 9 ​

Selesaikan sistem persamaan berikut dengan menggunakan matriks.

  

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I. Sutiawan

Master Teacher

Mahasiswa/Alumni Universitas Pasundan

Jawaban terverifikasi

Jawaban

.

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank dan end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank z end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table end style. 

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Pembahasan

Diketahui sistem persamaan berikut: Sehingga, Solusi persmaan tersebut dengan menggunakan metode Cramer adalah: Keterangan : deteminan dari matriks koefisien. : deteminan kofisien x yang diganti dengan konstanta. : deteminan kofisien x yang diganti dengan konstanta. : deteminan kofisien x yang diganti dengan konstanta. Tentukan dengan metode Minor-Kofaktor Misalnya kita pilih baris ke-1. Elemen-elemen baris ke-1 adalah: . Sehingga, Sehingga diperoleh: Jadi, .

Diketahui sistem persamaan berikut:

begin mathsize 14px style open curly brackets table attributes columnalign left end attributes row cell x plus 2 y minus z equals 8 end cell row cell 2 x minus y equals 0 end cell row cell 2 x minus y plus 3 z equals negative 9 end cell end table close end style 

Sehingga, 

 begin mathsize 14px style a subscript 11 equals 1 comma space a subscript 12 equals 2 comma space a subscript 13 equals negative 1 comma space c subscript 1 equals 8a subscript 21 equals 2 comma space a subscript 22 equals negative 1 comma space a subscript 23 equals 0 comma space c subscript 2 equals 0a subscript 31 equals 2 comma space a subscript 32 equals negative 1 comma space a subscript 33 equals 3 comma space c subscript 3 equals negative 9 end style 

Solusi persmaan tersebut dengan menggunakan metode Cramer adalah:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell straight D subscript x over straight D end cell row y equals cell straight D subscript y over straight D end cell row z equals cell straight D subscript z over straight D end cell end table end style  

Keterangan

  • straight D : deteminan dari matriks koefisien.
  • straight D subscript straight x : deteminan kofisien x yang diganti dengan konstanta.
  • straight D subscript straight y : deteminan kofisien x yang diganti dengan konstanta.
  • straight D subscript straight z : deteminan kofisien x yang diganti dengan konstanta. 

Tentukan begin mathsize 14px style straight D comma space straight D subscript x comma space straight D subscript y comma space dan space straight D subscript z end style dengan metode Minor-Kofaktor

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row straight D equals cell open vertical bar table row 1 2 cell negative 1 end cell row 2 cell negative 1 end cell 0 row 2 cell negative 1 end cell 3 end table close vertical bar end cell row cell straight D subscript x end cell equals cell open vertical bar table row 8 2 cell negative 1 end cell row 0 cell negative 1 end cell 0 row cell negative 9 end cell cell negative 1 end cell 3 end table close vertical bar end cell row cell straight D subscript y end cell equals cell open vertical bar table row 1 8 cell negative 1 end cell row 2 0 0 row 2 cell negative 9 end cell 3 end table close vertical bar end cell row cell straight D subscript z end cell equals cell open vertical bar table row 1 2 8 row 2 cell negative 1 end cell 0 row 2 cell negative 1 end cell cell negative 9 end cell end table close vertical bar end cell end table end style 

Misalnya kita pilih baris ke-1. Elemen-elemen baris ke-1 adalah: begin mathsize 14px style a subscript 11 comma space a subscript 12 comma space dan space a subscript 13 end style. Sehingga,

table attributes columnalign right center left columnspacing 0px end attributes row straight D equals cell begin inline style sum from i equals 1 to 3 of end style a subscript i j end subscript times straight C subscript i j end subscript end cell row blank equals cell a subscript 11 times straight C subscript 11 plus a subscript 12 times straight C subscript 12 plus a subscript 13 times straight C subscript 13 end cell row blank equals cell a subscript 11 times open parentheses negative 1 close parentheses to the power of 1 plus 1 end exponent times straight M subscript 11 plus a subscript 12 times open parentheses negative 1 close parentheses to the power of 1 plus 2 end exponent times straight M subscript 12 end cell row blank blank cell plus a subscript 13 times open parentheses negative 1 close parentheses to the power of 1 plus 3 end exponent times straight M subscript 13 end cell row blank equals cell 1 times 1 times open vertical bar table row cell negative 1 end cell 0 row cell negative 1 end cell 3 end table close vertical bar plus 2 times open parentheses negative 1 close parentheses times open vertical bar table row 2 0 row 2 3 end table close vertical bar end cell row blank blank cell plus open parentheses negative 1 close parentheses times 1 times open vertical bar table row 2 cell negative 1 end cell row 2 cell negative 1 end cell end table close vertical bar end cell row blank equals cell open parentheses open parentheses negative 1 close parentheses times 3 minus 0 times open parentheses negative 1 close parentheses close parentheses minus 2 times open parentheses 2 times 3 minus 0 times 2 close parentheses end cell row blank blank cell negative 1 times open parentheses 2 times open parentheses negative 1 close parentheses minus open parentheses negative 1 close parentheses times 2 close parentheses end cell row blank equals cell negative 3 minus 2 times 6 minus 1 times 0 end cell row blank equals cell negative 3 minus 12 minus 0 end cell row blank equals cell negative 15 end cell end table

table attributes columnalign right center left columnspacing 0px end attributes row cell straight D subscript x end cell equals cell begin inline style sum from i equals 1 to 3 of end style a subscript i j end subscript times straight C subscript i j end subscript end cell row blank equals cell a subscript 11 times straight C subscript 11 plus a subscript 12 times straight C subscript 12 plus a subscript 13 times straight C subscript 13 end cell row blank equals cell a subscript 11 times open parentheses negative 1 close parentheses to the power of 1 plus 1 end exponent times straight M subscript 11 plus a subscript 12 times open parentheses negative 1 close parentheses to the power of 1 plus 2 end exponent times straight M subscript 12 end cell row blank blank cell plus a subscript 13 times open parentheses negative 1 close parentheses to the power of 1 plus 3 end exponent times straight M subscript 13 end cell row blank equals cell 8 times 1 times open vertical bar table row cell negative 1 end cell 0 row cell negative 1 end cell 3 end table close vertical bar plus 2 times open parentheses negative 1 close parentheses times open vertical bar table row 0 0 row cell negative 9 end cell 3 end table close vertical bar end cell row blank blank cell plus open parentheses negative 1 close parentheses times 1 times open vertical bar table row 0 cell negative 1 end cell row cell negative 9 end cell cell negative 1 end cell end table close vertical bar end cell row blank equals cell 8 times open parentheses open parentheses negative 1 close parentheses times 3 minus 0 times open parentheses negative 1 close parentheses close parentheses minus 2 times open parentheses 0 times 3 minus 0 times open parentheses negative 9 close parentheses close parentheses end cell row blank blank cell negative 1 times open parentheses 0 times open parentheses negative 1 close parentheses minus open parentheses negative 1 close parentheses times open parentheses negative 9 close parentheses close parentheses end cell row blank equals cell 8 times open parentheses negative 3 close parentheses minus 2 times 0 minus 1 times open parentheses negative 9 close parentheses end cell row blank equals cell negative 24 minus 0 plus 9 end cell row blank equals cell negative 15 end cell end table  

table attributes columnalign right center left columnspacing 0px end attributes row cell straight D subscript y end cell equals cell begin inline style sum from i equals 1 to 3 of end style a subscript i j end subscript times straight C subscript i j end subscript end cell row blank equals cell a subscript 11 times straight C subscript 11 plus a subscript 12 times straight C subscript 12 plus a subscript 13 times straight C subscript 13 end cell row blank equals cell a subscript 11 times open parentheses negative 1 close parentheses to the power of 1 plus 1 end exponent times straight M subscript 11 plus a subscript 12 times open parentheses negative 1 close parentheses to the power of 1 plus 2 end exponent times straight M subscript 12 end cell row blank blank cell plus a subscript 13 times open parentheses negative 1 close parentheses to the power of 1 plus 3 end exponent times straight M subscript 13 end cell row blank equals cell 1 times 1 times open vertical bar table row 0 0 row cell negative 9 end cell 3 end table close vertical bar plus 8 times open parentheses negative 1 close parentheses times open vertical bar table row 2 0 row 2 3 end table close vertical bar end cell row blank blank cell plus open parentheses negative 1 close parentheses times 1 times open vertical bar table row 2 0 row 2 cell negative 9 end cell end table close vertical bar end cell row blank equals cell open parentheses 0 times 3 minus 0 times open parentheses negative 9 close parentheses close parentheses minus 8 times open parentheses 2 times 3 minus 0 times 2 close parentheses end cell row blank blank cell negative 1 times open parentheses 2 times open parentheses negative 9 close parentheses minus 0 times 2 close parentheses end cell row blank equals cell 0 minus 8 times 6 minus 1 times open parentheses negative 18 close parentheses end cell row blank equals cell negative 48 plus 18 end cell row blank equals cell negative 30 end cell end table  

table attributes columnalign right center left columnspacing 0px end attributes row cell straight D subscript z end cell equals cell begin inline style sum from i equals 1 to 3 of end style a subscript i j end subscript times straight C subscript i j end subscript end cell row blank equals cell a subscript 11 times straight C subscript 11 plus a subscript 12 times straight C subscript 12 plus a subscript 13 times straight C subscript 13 end cell row blank equals cell a subscript 11 times open parentheses negative 1 close parentheses to the power of 1 plus 1 end exponent times straight M subscript 11 plus a subscript 12 times open parentheses negative 1 close parentheses to the power of 1 plus 2 end exponent times straight M subscript 12 end cell row blank blank cell plus a subscript 13 times open parentheses negative 1 close parentheses to the power of 1 plus 3 end exponent times straight M subscript 13 end cell row blank equals cell 1 times 1 times open vertical bar table row cell negative 1 end cell 0 row cell negative 1 end cell cell negative 9 end cell end table close vertical bar plus 2 times open parentheses negative 1 close parentheses times open vertical bar table row 2 0 row 2 cell negative 9 end cell end table close vertical bar end cell row blank blank cell plus 8 times 1 times open vertical bar table row 2 cell negative 1 end cell row 2 cell negative 1 end cell end table close vertical bar end cell row blank equals cell open parentheses open parentheses negative 1 close parentheses times open parentheses negative 9 close parentheses minus 0 times open parentheses negative 1 close parentheses close parentheses minus 2 times open parentheses 2 times open parentheses negative 9 close parentheses minus 0 times 2 close parentheses end cell row blank blank cell plus 8 times open parentheses 2 times open parentheses negative 1 close parentheses minus open parentheses negative 1 close parentheses times 2 close parentheses end cell row blank equals cell 9 minus 2 times open parentheses negative 18 close parentheses plus 8 times 0 end cell row blank equals cell 9 plus 36 plus 0 end cell row blank equals 45 end table  

Sehingga diperoleh:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell straight D subscript x over straight D end cell row blank equals cell fraction numerator negative 15 over denominator negative 15 end fraction end cell row blank equals 1 row y equals cell straight D subscript y over straight D end cell row blank equals cell fraction numerator negative 30 over denominator negative 15 end fraction end cell row blank equals 2 row z equals cell straight D subscript z over straight D end cell row blank equals cell fraction numerator 45 over denominator negative 15 end fraction end cell row blank equals cell negative 3 end cell end table end style 

Jadi, begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank y end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank dan end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank z end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table end style. 

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