Matrices — Class 12 Maths Solution

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We have, $\cos \theta \left[ {\begin{array}{cccccccccccccccccccc}{\cos \theta }&{\sin \theta }\\{ - \sin \theta }&{\cos \theta }\end{array}} \right] + \sin \theta \left[ {\begin{array}{cccccccccccccccccccc}{\sin \theta }&{ - \cos \theta }\\{\cos \theta }&{\sin \theta }\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}{{{\cos }^2}\theta }&{\sin \theta \cos \theta }\\{ - \sin \theta \cos \theta }&{{{\cos }^2}\theta }\end{array}} \right] + \left[ {\begin{array}{cccccccccccccccccccc}{{{\sin }^2}\theta }&{ - \sin \theta \cos \theta }\\{\sin \theta \cos \theta }&{{{\sin }^2}\theta }\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}{{{\cos }^2}\theta + {{\sin }^2}\theta }&{\sin \theta \cos \theta - \sin \theta \cos \theta }\\{ - \sin \theta \cos \theta + \cos \theta \sin \theta }&{{{\cos }^2}\theta + {{\sin }^2}\theta }\end{array}} \right]$

$= \left[ {\begin{array}{cccccccccccccccccccc}1&0\\0&1\end{array}} \right]$