Diketahui matriks M=(cos xk1 sin xk sin x−cos x) dengan k konstanta bukan nol. Tentukan transpose dari M2.

Pertanyaan

Diketahui matriks $M=\begin{pmatrix}\cos\;x&k\;\sin\;x\\\frac1k\;\sin\;x&-\cos\;x\end{pmatrix}$ dengan $k$ konstanta bukan nol. Tentukan transpose dari $M^2$ .

S. Ayu

Master Teacher

Mahasiswa/Alumni Universitas Muhammadiyah Prof. DR. Hamka

Jawaban terverifikasi

Jawaban

transpose dari M squared

adalah

open parentheses table row 1 0 row 0 1 end table close parentheses

Pembahasan

Perkalian kuadarat matriks 2 cross times 2 yaitu:

open parentheses table row a b row c d end table close parentheses open parentheses table row a b row c d end table close parentheses equals open parentheses table row cell a times a plus b times c end cell cell a times b plus b times d end cell row cell c times a plus d times c end cell cell c times b plus d times d end cell end table close parentheses

Pada identitas trigonometri, berlaku:

sin squared space x plus cos squared space x equals 1

Penyelesaiannya yaitu:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row M equals cell open parentheses table row cell cos space x end cell cell k space sin space x end cell row cell 1 over k space sin space x end cell cell negative cos space x end cell end table close parentheses end cell row cell M squared end cell equals cell open parentheses table row cell cos space x end cell cell k space sin space x end cell row cell 1 over k space sin space x end cell cell negative cos space x end cell end table close parentheses squared end cell row blank equals cell open parentheses table row cell cos space x end cell cell k space sin space x end cell row cell 1 over k space sin space x end cell cell negative cos space x end cell end table close parentheses open parentheses table row cell cos space x end cell cell k space sin space x end cell row cell 1 over k space sin space x end cell cell negative cos space x end cell end table close parentheses end cell row blank equals cell open parentheses table row cell cos space x open parentheses cos space x close parentheses plus down diagonal strike k space sin space x open parentheses fraction numerator 1 over denominator down diagonal strike k end fraction space sin space x close parentheses end cell cell cos space x open parentheses k space sin space x close parentheses plus k space sin space x open parentheses negative cos space x close parentheses end cell row cell 1 over k space sin space x open parentheses cos space x close parentheses plus open parentheses negative cos space x close parentheses open parentheses 1 over k space sin space x close parentheses end cell cell fraction numerator 1 over denominator down diagonal strike k end fraction space sin space x open parentheses down diagonal strike k space sin space x close parentheses plus open parentheses negative cos space x close parentheses open parentheses negative cos space x close parentheses end cell end table close parentheses end cell row blank equals cell open parentheses table row cell cos squared space x plus sin squared space x end cell cell k space cos space x times sin space x minus k space sin space x times cos space x end cell row cell fraction numerator sin space x times cos space x over denominator k end fraction minus fraction numerator cos space x times sin space x over denominator k end fraction end cell cell sin squared space x plus cos squared space x end cell end table close parentheses end cell row blank equals cell open parentheses table row 1 0 row 0 1 end table close parentheses end cell end table end style$

Kemudian tentukan transpose dari matriks M squared .

Rumus transpose 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 T end cell equals cell open parentheses table row a c row b d end table close parentheses end cell end table

Diperoleh hasilnya:

table attributes columnalign right center left columnspacing 0px end attributes row cell M squared end cell equals cell open parentheses table row 1 0 row 0 1 end table close parentheses end cell row cell open parentheses M squared close parentheses to the power of T end cell equals cell open parentheses table row 1 0 row 0 1 end table close parentheses end cell end table

Dengan demikian, transpose dari M squared adalah open parentheses table row 1 0 row 0 1 end table close parentheses .