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  4. A= [cos θ&-sin θ sin θ&cos θ] and AB= BA =I , then B is equal to

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Q. $ A= \begin{bmatrix}cos\,\theta&-sin\,\theta\\ sin\,\theta&cos\,\theta\end{bmatrix} $ and $ AB= BA =I $ , then $ B $ is equal to

MHT CETMHT CET 2009

Solution:

Given, $A=\begin{bmatrix}cos \,\theta&-sin \,\theta\\ sin \,\theta&cos \, \theta\end{bmatrix}$
and $AB = BA = I$
$\Rightarrow B = A^{-1} I = A^{-1}$
$ =\frac{1}{cos^{2} \theta +sin ^{2}\theta }\begin{bmatrix}cos\, \theta&sin\, \theta\\ -sin \,\theta&cos \,\theta\end{bmatrix}$
$ \Rightarrow B=\begin{bmatrix}cos\, \theta&sin\, \theta\\ -sin \,\theta&cos \,\theta\end{bmatrix}$