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Question

Simplify: $$$\cos\theta\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix} + \sin\theta\begin{bmatrix}\sin\theta&-\cos\theta\\ \cos\theta&\sin\theta\end{bmatrix}$

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Answer

Given: $\cos\theta\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix} + \sin\theta\begin{bmatrix}\sin\theta&-\cos\theta \\ \cos\theta&\ \sin\theta\end{bmatrix}$
$=\begin{bmatrix}\cos^2\theta&\cos\theta\sin\theta\\-\sin\theta\cos\theta&\cos^2\theta\end{bmatrix} + \begin{bmatrix}\sin^2\theta&-\cos\theta\sin\theta\\ \cos\theta\sin\theta&\ \sin^2\theta\end{bmatrix} = \begin{bmatrix}1&0\\0&1\end{bmatrix}$

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