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Diketahui matriks A=[32​05​], B=[xy​−11​] dan C=[0−15​−15​]. AT adalah transpose dari A. Jika A⊤⋅B=C tentukan nilai x dan y!

Pertanyaan

Diketahui matriks straight A equals open square brackets table row 3 0 row 2 5 end table close square brackets comma space straight B equals open square brackets table row straight x cell negative 1 end cell row straight y 1 end table close square brackets space dan space straight C equals open square brackets table row 0 cell negative 1 end cell row cell negative 15 end cell 5 end table close square bracketsstraight A to the power of straight T adalah transpose dari A. Jika straight A to the power of ⊤ times straight B equals straight C tentukan nilai x space dan space y!

S. Difhayanti

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. Dr. Hamka

Jawaban terverifikasi

Jawaban

nilai x space dan space y berturut-turut adalah 2 space dan space minus 3.

Pembahasan

Ingat transpose matriks.

table attributes columnalign right center left columnspacing 0px end attributes row straight A equals cell open parentheses table row straight a straight b row straight c straight d end table close parentheses end cell row cell straight A to the power of straight T end cell equals cell open parentheses table row straight a straight c row straight b straight d end table close parentheses end cell end table

Diketahui: 

straight A equals open square brackets table row 3 0 row 2 5 end table close square brackets comma space straight B equals open square brackets table row straight x cell negative 1 end cell row straight y 1 end table close square brackets space dan space straight C equals open square brackets table row 0 cell negative 1 end cell row cell negative 15 end cell 5 end table close square brackets

Mencari kesamaan matriks straight A to the power of ⊤ times straight B equals straight C:

table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses table row 3 0 row 2 5 end table close parentheses to the power of straight T open parentheses table row x cell negative 1 end cell row y 1 end table close parentheses end cell equals cell open parentheses table row 0 cell negative 1 end cell row cell negative 15 end cell 5 end table close parentheses end cell row cell open parentheses table row 3 2 row 0 5 end table close parentheses open parentheses table row x cell negative 1 end cell row y 1 end table close parentheses end cell equals cell open parentheses table row 0 cell negative 1 end cell row cell negative 15 end cell 5 end table close parentheses end cell row cell open parentheses table row cell 3 x plus 2 y end cell cell 3 open parentheses negative 1 close parentheses plus 2 times 1 end cell row cell 0 times x plus 5 y end cell cell 0 times open parentheses negative 1 close parentheses plus 5 times 1 end cell end table close parentheses end cell equals cell open parentheses table row 0 cell negative 1 end cell row cell negative 15 end cell 5 end table close parentheses end cell row cell open parentheses table row cell 3 x plus 2 y end cell cell negative 1 end cell row cell 5 y end cell 5 end table close parentheses end cell equals cell open parentheses table row 0 cell negative 1 end cell row cell negative 15 end cell 5 end table close parentheses end cell end table 

Mencari nilai x space dan space y:

table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x plus 2 y end cell equals 0 row cell 5 y end cell equals cell negative 15 end cell row cell 5 y end cell equals cell negative 15 end cell row y equals cell fraction numerator negative 15 over denominator 5 end fraction end cell row y equals cell negative 3 end cell row blank blank blank row cell 3 x plus 2 y end cell equals 0 row cell 3 x plus 2 open parentheses negative 3 close parentheses end cell equals 0 row cell 3 x minus 6 end cell equals 0 row cell 3 x end cell equals 6 row x equals 2 end table 

Jadi, nilai x space dan space y berturut-turut adalah 2 space dan space minus 3.

311

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