Question

If $F\left(\alpha \right)=\left(\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)$ .
find ${\left\{F\right(x\left)\right\}}^{-1}=?$

• A. $F(-x)$
• B. $-F\left(x\right)$
• C. $F\left(1/x\right)$
• D. $1/F\left(x\right)$

Answer 1

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

$F\left(\alpha \right)=\left(\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right)$
$F\left(x\right)=\left(\begin{array}{ccc}\cos x& -\sin x& 0\\ \sin x& \cos x& 0\\ 0& 0& 1\end{array}\right)$
$|F\left(x\right)|=1$
$adjF\left(x\right)=\left(\begin{array}{ccc}\cos x& \sin x& 0\\ -\sin x& \cos x& 0\\ 0& 0& 1\end{array}\right)$
${\left(F\right(x\left)\right)}^{-1}=\left(\begin{array}{ccc}\cos x& \sin x& 0\\ -\sin x& \cos x& 0\\ 0& 0& 1\end{array}\right)=F(-x)$
$\therefore {\left(F\right(x\left)\right)}^{-1}=F(-x)$
Hence, optiono A.