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1. Diketahui matriks  maka tentukan a. Determinan   b. Matriks kofaktor c. Invers matriks

Pertanyaan

1. Diketahui matriks begin mathsize 14px style M equals open square brackets table row 1 2 3 row 2 5 3 row 1 0 8 end table close square brackets end style maka tentukan

a. Determinan undefined 

b. Matriks kofaktor undefined

c. Invers matriks undefined undefined 

  1. ...undefined 

  2. ...undefined 

Pembahasan Video:

Pembahasan Soal:

a. Determinan begin mathsize 14px style M end style

Determinan undefined dapat dicari dengan menggunakan aturan Sarrus. Agar lebih mudah, kita tulis kembali elemen-elemen pada kolom ke-begin mathsize 14px style 1 end style dan ke-undefined di sebelah kanan matriks undefined sebagai berikut.

begin mathsize 14px style M equals open vertical bar table row 1 2 3 row 2 5 3 row 1 0 8 end table close vertical bar table row 1 2 row 2 5 row 1 0 end table end style

Akibatnya, bisa dicari nilai determinan undefined dengan menggunakan aturan Sarrus sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell det space M end cell equals cell a subscript 11 a subscript 22 a subscript 33 plus a subscript 12 a subscript 23 a subscript 31 plus a subscript 13 a subscript 21 a subscript 32 minus a subscript 13 a subscript 22 a subscript 31 minus a subscript 11 a subscript 23 a subscript 32 minus a subscript 12 a subscript 21 a subscript 33 end cell row blank equals cell open parentheses 1 times 5 times 8 close parentheses plus open parentheses 2 times 3 times 1 close parentheses plus open parentheses 3 times 2 times 0 close parentheses minus open parentheses 3 times 5 times 1 close parentheses minus open parentheses 1 times 3 times 0 close parentheses minus open parentheses 2 times 2 times 8 close parentheses end cell row blank equals cell 40 plus 6 plus 0 minus 15 minus 0 minus 32 end cell row blank equals cell 46 minus 47 end cell row blank equals cell negative 1 end cell end table end style

Dengan demikian, nilai determinan undefined adalah begin mathsize 14px style negative 1 end style.
 

b. Matriks Kofaktor undefined

Dengan menggunakan metode minor-kofaktor, kita bisa mencari matriks kofaktor undefined. Misalkan, begin mathsize 14px style A subscript i j end subscript end style merupakan matriks bagian dari matriks begin mathsize 14px style A end style yang diperoleh dengan cara menghilangkan baris ke-begin mathsize 14px style i end style dan kolom ke-begin mathsize 14px style j end style. Akibatnya, bisa dicari minor matriks undefined (diberi notasi begin mathsize 14px style M subscript i j end subscript end style) dan kofaktor matriks undefined (diberi notasi begin mathsize 14px style C subscript i j end subscript end style) sebagai berikut.

begin mathsize 14px style M subscript i j end subscript equals d e t space A subscript i j end subscript end style 

dan

begin mathsize 14px style C subscript i j end subscript equals open parentheses negative 1 close parentheses to the power of i plus j end exponent M subscript i j end subscript end style 

Oleh karena itu, akan kita cari kofaktor dari masing-masing baris dan kolom pada matriks undefined sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 11 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 1 plus 1 end exponent open vertical bar table row 5 3 row 0 8 end table close vertical bar end cell row blank equals cell left parenthesis negative 1 right parenthesis squared times left parenthesis 40 minus 0 right parenthesis end cell row blank equals cell 1 times 40 end cell row blank equals 40 end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 12 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 1 plus 2 end exponent open vertical bar table row cell 2 end cell cell 3 end cell row cell 1 end cell cell 8 end cell end table close vertical bar space end cell row blank equals cell left parenthesis negative 1 right parenthesis cubed times left parenthesis 16 minus 3 right parenthesis space end cell row blank equals cell negative 1 times 13 space end cell row blank equals cell negative 13 end cell end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 13 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 1 plus 3 end exponent open vertical bar table row 2 5 row 1 0 end table close vertical bar space end cell row blank equals cell left parenthesis negative 1 right parenthesis to the power of 4 times left parenthesis 0 minus 5 right parenthesis space end cell row blank equals cell 1 times left parenthesis negative 5 right parenthesis space end cell row blank equals cell negative 5 end cell end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 21 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 2 plus 1 end exponent open vertical bar table row 2 3 row 0 8 end table close vertical bar space end cell row blank equals cell left parenthesis negative 1 right parenthesis cubed times left parenthesis 16 minus 0 right parenthesis space end cell row blank equals cell negative 1 times 16 space end cell row blank equals cell negative 16 end cell end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 22 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 2 plus 2 end exponent open vertical bar table row cell 1 end cell cell 3 end cell row cell 1 end cell cell 8 end cell end table close vertical bar space space end cell row blank equals cell left parenthesis negative 1 right parenthesis to the power of 4 times left parenthesis 8 minus 3 right parenthesis space end cell row blank equals cell 1 times 5 space end cell row blank equals 5 end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 23 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 2 plus 3 end exponent open vertical bar table row 1 2 row 1 0 end table close vertical bar space space end cell row blank equals cell left parenthesis negative 1 right parenthesis to the power of 5 times left parenthesis 0 minus 2 right parenthesis space end cell row blank equals cell negative 1 times left parenthesis negative 2 right parenthesis space end cell row blank equals 2 end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 31 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 3 plus 1 end exponent open vertical bar table row 2 3 row 5 3 end table close vertical bar space space end cell row blank equals cell left parenthesis negative 1 right parenthesis to the power of 4 times left parenthesis 6 minus 15 right parenthesis space end cell row blank equals cell 1 times left parenthesis negative 9 right parenthesis end cell row blank equals cell negative 9 end cell end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 32 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 3 plus 2 end exponent open vertical bar table row 1 3 row 2 3 end table close vertical bar space space end cell row blank equals cell left parenthesis negative 1 right parenthesis to the power of 5 times left parenthesis 3 minus 6 right parenthesis space end cell row blank equals cell negative 1 times left parenthesis negative 3 right parenthesis space end cell row blank equals 3 end table end style

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell C subscript 33 end cell equals cell left parenthesis negative 1 right parenthesis to the power of 3 plus 3 end exponent open vertical bar table row 1 2 row 2 5 end table close vertical bar space space end cell row blank equals cell left parenthesis negative 1 right parenthesis to the power of 6 times left parenthesis 5 minus 4 right parenthesis space end cell row blank equals cell 1 times 1 space end cell row blank equals 1 end table end style

Dengan demikian, bisa kita susun matriks kofaktor undefined sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell k o f space M end cell equals cell open square brackets table row cell C subscript 11 end cell cell C subscript 12 end cell cell C subscript 13 end cell row cell C subscript 21 end cell cell C subscript 22 end cell cell C subscript 23 end cell row cell C subscript 31 end cell cell C subscript 32 end cell cell C subscript 33 end cell end table close square brackets end cell row blank equals cell open square brackets table row 40 cell negative 13 end cell cell negative 5 end cell row cell negative 16 end cell 5 2 row cell negative 9 end cell 3 1 end table close square brackets end cell end table end style
 

c. Invers Matriks undefined

Dengan menggunakan rumus invers matriks undefined dengan Adjoin berikut

begin mathsize 14px style M to the power of negative 1 end exponent equals fraction numerator 1 over denominator d e t space M end fraction a d j left parenthesis M right parenthesis end style

akibatnya, bisa dicari invers matriks undefined.

Adjoin matriks merupakan transpose dari matriks kofaktor. Oleh karena itu, Adjoin matriks undefined bisa ditentukan sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a d j left parenthesis M right parenthesis end cell equals cell open parentheses k o f left parenthesis M right parenthesis close parentheses to the power of T end cell row blank equals cell open parentheses open square brackets table row 40 cell negative 13 end cell cell negative 5 end cell row cell negative 16 end cell 5 2 row cell negative 9 end cell 3 1 end table close square brackets close parentheses to the power of T end cell row blank equals cell open square brackets table row 40 cell negative 16 end cell cell negative 9 end cell row cell negative 13 end cell 5 3 row cell negative 5 end cell 2 1 end table close square brackets end cell end table end style

Dengan demikian, bisa ditentukan invers matriks undefined adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank blank row cell M to the power of negative 1 end exponent end cell equals cell fraction numerator 1 over denominator d e t space M end fraction a d j left parenthesis M right parenthesis end cell row blank equals cell fraction numerator 1 over denominator negative 1 end fraction open square brackets table row 40 cell negative 16 end cell cell negative 9 end cell row cell negative 13 end cell 5 3 row cell negative 5 end cell 2 1 end table close square brackets end cell row blank equals cell negative 1 open square brackets table row 40 cell negative 16 end cell cell negative 9 end cell row cell negative 13 end cell 5 3 row cell negative 5 end cell 2 1 end table close square brackets end cell row blank equals cell open square brackets table row cell negative 40 end cell 16 9 row 13 cell negative 5 end cell cell negative 3 end cell row 5 cell negative 2 end cell cell negative 1 end cell end table close square brackets end cell end table end style 

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Terakhir diupdate 05 Mei 2021

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