Question

Simplify \(\cos\theta\) \(\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}\)+\(\sin\theta\) \(\begin{bmatrix}\sin\theta&-\cos\theta\\\cos\theta&\sin\theta\end{bmatrix}\)

Answer 1

\(\cos\theta\)\(\begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix}\)+\(\sin\theta\)\(\begin{bmatrix}\sin\theta&-\cos\theta\\\cos\theta&\sin\theta\end{bmatrix}\)

=\(\begin{bmatrix}\cos^2\theta&\cos\theta\sin\theta\\-\sin\theta\cos\theta&\cos^2\theta\end{bmatrix}\)+\(\begin{bmatrix}\sin^2\theta&-\sin\theta\cos\theta\\\sin\theta\cos\theta&\sin^2\theta\end{bmatrix}\)

=\(\begin{bmatrix}\cos^2\theta+\sin^2\theta&\cos\theta\sin\theta-\sin\theta\cos\theta\\-\sin\theta\cos\theta+\sin\theta\cos\theta&\cos^2\theta+\sin^2\theta\end{bmatrix}\)

=\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)