Question

Simplify : $\cos Q\left[\begin{array}{cc}\cos Q& \sin Q\\ -\sin Q& \cos Q\end{array}\right]+\sin Q\left[\begin{array}{cc}\sin Q& -\cos Q\\ \cos Q& \sin Q\end{array}\right]$

Answer 1

$\cos Q\left[\begin{array}{cc}\cos Q& \sin Q\\ -\sin Q& \cos Q\end{array}\right]+\sin Q\left[\begin{array}{cc}\sin Q& -\cos Q\\ \cos Q& \sin Q\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}{\cos }^{2}Q& \sin Q\cos Q\\ -\sin Q\cos Q& {\cos }^{2}Q\end{array}\right]+\left[\begin{array}{cc}{\sin }^{2}Q& -\sin Q\cos Q\\ \sin Q\cos Q& {\sin }^{2}Q\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}{\cos }^{2}Q+{\sin }^{2}Q& \sin Q\cos Q-\sin Q\cos Q\\ -\sin Q\cos Q+\sin Q\cos Q& {\cos }^{2}Q+{\sin }^{2}Q\end{array}\right]$

$\Rightarrow \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

$\Rightarrow I$

Hence, solved.