Let F( ) = [ * 20 c & - &0 & &0 0&0&1 ], where R. Then [F( )]^ - … - Vidyadip

MCQ

Let $F(\alpha ) = \left[ {\begin{array}{*{20}{c}}{\cos \alpha }&{ - \sin \alpha }&0\\{\sin \alpha }&{\cos \alpha }&0\\0&0&1\end{array}} \right]$, where $\alpha \in R.$ Then ${[F(\alpha )]^{ - 1}}$ is equal to

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

✓

Answer

Correct option: A.

$F( - \alpha )$

a
(a) We have

$F(\alpha )\,F( - \alpha )\, = \,\left[ {\begin{array}{*{20}{c}}{\cos \alpha }&{ - \sin \alpha }&0\\{\sin \alpha }&{\cos \alpha }&0\\0&0&1\end{array}} \right]\,\,\left[ {\begin{array}{*{20}{c}}{\cos \alpha }&{\sin \alpha }&0\\{ - \sin \alpha }&{\cos \alpha }&0\\0&0&1\end{array}} \right]$

= $\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right] = I$

$\therefore $ $F( - \alpha ) = {[F(\alpha )]^{ - 1}}$.

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