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Pertanyaan

Sebuah matriks dikatakan orthogonal jika begin mathsize 14px style straight A to the power of negative 1 end exponent equals straight A to the power of straight T. end style Diketahui begin mathsize 14px style straight A equals open parentheses table row cell straight x plus straight y end cell 0 row 0 cell straight x minus straight y end cell end table close parentheses end style adalah matriks orhogonal, nilai dari begin mathsize 14px style straight x squared plus straight y squared equals end style ....

  1. -1

  2. 0

  3. 1

  4. 2

  5. 4

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C.

Pembahasan

Diketahui A adalah matriks orthogonal, maka begin mathsize 14px style straight A to the power of negative 1 end exponent equals straight A to the power of straight T. end style 

Perhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell AA to the power of negative 1 end exponent end cell equals straight I row cell AA to the power of straight T end cell equals straight I end table end style 

dan

begin mathsize 14px style straight A to the power of straight T equals open parentheses table row cell straight x plus straight y end cell 0 row 0 cell straight x minus straight y end cell end table close parentheses end style 

Sehingga kita peroleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell AA to the power of straight T end cell equals straight I row cell open parentheses table row cell straight x plus straight y end cell 0 row 0 cell straight x minus straight y end cell end table close parentheses    open parentheses table row cell straight x plus straight y end cell 0 row 0 cell straight x minus straight y end cell end table close parentheses end cell equals cell open parentheses table row 1 0 row 0 1 end table close parentheses end cell row cell open parentheses table row cell open parentheses straight x plus straight y close parentheses squared end cell 0 row 0 cell open parentheses straight x minus straight y close parentheses squared end cell end table close parentheses end cell equals cell open parentheses table row 1 0 row 0 1 end table close parentheses end cell row blank blank blank end table end style 

Dari kesamaan matriks di atas, kita peroleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses straight x plus straight y close parentheses squared end cell equals 1 row cell straight x squared plus 2 xy plus straight y squared end cell equals 1 row blank blank blank row cell open parentheses straight x minus straight y close parentheses squared end cell equals 1 row cell straight x squared minus 2 xy plus straight y squared end cell equals 1 row blank blank blank end table end style 

Maka,

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 1 plus 1 end cell equals cell open parentheses straight x squared plus 2 xy plus straight y squared close parentheses plus open parentheses straight x squared minus 2 xy plus straight y squared close parentheses end cell row 2 equals cell 2 straight x squared plus 2 straight y squared end cell row cell straight x squared plus straight y squared end cell equals 1 end table end style 

Jadi, jawaban yang tepat adalah C.

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