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Pertanyaan

Diketahui X adalah matriks berordo 2 cross times 2. Tentukan X squared pada persamaan matriks berikut.

open parentheses table row 4 cell negative 3 end cell row 1 2 end table close parentheses X equals open parentheses table row 20 cell negative 5 end cell row cell negative 6 end cell 7 end table close parentheses

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

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Pembahasan

Pada persamaan matriks A X equals B berlaku:

table attributes columnalign right center left columnspacing 0px end attributes row cell A X end cell equals B row cell A to the power of negative 1 end exponent A X end cell equals cell A to the power of negative 1 end exponent B end cell row cell I X end cell equals cell A to the power of negative 1 end exponent B end cell row X equals cell A to the power of negative 1 end exponent B 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 open vertical bar A close vertical bar 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 d minus b 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

Perkalian dua buah matriks 2 cross times 2 berlaku:

open parentheses table row a b row c d end table close parentheses open parentheses table row p q row r s end table close parentheses equals open parentheses table row cell a times p plus b times r end cell cell a times q plus b times s end cell row cell c times p plus d times r end cell cell c times q plus d times s end cell end table close parentheses

Diperoleh penyelesainnya yaitu:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row 4 cell negative 3 end cell row 1 2 end table close parentheses X end cell equals cell open parentheses table row 20 cell negative 5 end cell row cell negative 6 end cell 7 end table close parentheses end cell row X equals cell open parentheses table row 4 cell negative 3 end cell row 1 2 end table close parentheses to the power of negative 1 end exponent times open parentheses table row 20 cell negative 5 end cell row cell negative 6 end cell 7 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 4 open parentheses 2 close parentheses minus open parentheses negative 3 close parentheses open parentheses 1 close parentheses end fraction open parentheses table row 2 3 row cell negative 1 end cell 4 end table close parentheses open parentheses table row 20 cell negative 5 end cell row cell negative 6 end cell 7 end table close parentheses end cell row blank equals cell fraction numerator 1 over denominator 8 plus 3 end fraction open parentheses table row cell 2 open parentheses 20 close parentheses plus 3 open parentheses negative 6 close parentheses end cell cell 2 open parentheses negative 5 close parentheses plus 3 open parentheses 7 close parentheses end cell row cell negative 1 open parentheses 20 close parentheses plus 4 open parentheses negative 6 close parentheses end cell cell negative 1 open parentheses negative 5 close parentheses plus 4 open parentheses 7 close parentheses end cell end table close parentheses end cell row blank equals cell 1 over 11 open parentheses table row cell 40 minus 18 end cell cell negative 10 plus 21 end cell row cell negative 20 minus 24 end cell cell 5 plus 28 end cell end table close parentheses end cell row blank equals cell 1 over 11 open parentheses table row 22 11 row cell negative 44 end cell 33 end table close parentheses end cell row blank equals cell open parentheses table row cell 1 over 11 times 22 end cell cell 1 over 11 times 11 end cell row cell 1 over 11 times open parentheses negative 44 close parentheses end cell cell 1 over 11 times 33 end cell end table close parentheses end cell row blank equals cell open parentheses table row 2 1 row cell negative 4 end cell 3 end table close parentheses end cell end table

Kemudian tentukan matriks dari X squared seperti berikut:

table attributes columnalign right center left columnspacing 0px end attributes row X equals cell open parentheses table row 2 1 row cell negative 4 end cell 3 end table close parentheses end cell row cell X squared end cell equals cell open parentheses table row 2 1 row cell negative 4 end cell 3 end table close parentheses squared end cell row blank equals cell open parentheses table row 2 1 row cell negative 4 end cell 3 end table close parentheses open parentheses table row 2 1 row cell negative 4 end cell 3 end table close parentheses end cell row blank equals cell open parentheses table row cell 2 open parentheses 2 close parentheses plus 1 open parentheses negative 4 close parentheses end cell cell 2 open parentheses 1 close parentheses plus 1 open parentheses 3 close parentheses end cell row cell negative 4 open parentheses 2 close parentheses plus 3 open parentheses negative 4 close parentheses end cell cell negative 4 open parentheses 1 close parentheses plus 3 open parentheses 3 close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell 4 minus 4 end cell cell 2 plus 3 end cell row cell negative 8 minus 12 end cell cell negative 4 plus 9 end cell end table close parentheses end cell row blank equals cell open parentheses table row 0 5 row cell negative 20 end cell 5 end table close parentheses end cell end table

Dengan demikian, X squared pada persamaan matriks tersebut adalah open parentheses table row 0 5 row cell negative 20 end cell 5 end table close parentheses.

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