Let F( )= [ ccc & - & 0 & & 0 0 & 0 & 1 ], where R. Then [ F ( )]… - Vidyadip

MCQ

Let $F(\alpha)=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$, where $\alpha \in R$.
Then $[ F (\alpha)]^{-1}$ is equal to

✓
$F (-\alpha)$
B
$F \left(\alpha^{-1}\right)$
C
$F (2 \alpha)$
D
None of these

✓

Answer

Correct option: A.

$F (-\alpha)$

(A) $F(\alpha) \cdot F(-\alpha)$
$=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$
$=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]= I$
$\therefore[ F (\alpha)]^{-1}= F (-\alpha)$

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