Question

If $A=\left[\begin{array}{ccc}\cos x& \sin x& 0\\ -\sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]=f\left(x\right)$, then $A^{-1}$ =

• A. $f(-x)$
• B. $f\left(x\right)$
• C. $-f\left(x\right)$
• D. $-f(-x)$

Answer 1

The correct option is A $f(-x)$

$A=\left[\begin{array}{ccc}\cos x& \sin x& 0\\ -\sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]=f\left(x\right)$
$\Rightarrow |A|=1$
$Co-factormatrixis\left[\begin{array}{ccc}\cos x& \sin x& 0\\ -\sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$
$adjA=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$
$\therefore A^{-1}=\left[\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]=f(-x)$