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Mathematics Question on Matrices

Simplify cos⁡θ\cos\theta [cos⁡θsin⁡θ−sin⁡θcos⁡θ]\begin{bmatrix}\cos\theta&\sin\theta\\\\-\sin\theta&\cos\theta\end{bmatrix}+sin⁡θ\sin\theta [sin⁡θ−cos⁡θcos⁡θsin⁡θ]\begin{bmatrix}\sin\theta&-\cos\theta\\\\\cos\theta&\sin\theta\end{bmatrix}

Answer

\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}

=[cos⁡2θcos⁡θsin⁡θ−sin⁡θcos⁡θcos⁡2θ]\begin{bmatrix}\cos^2\theta&\cos\theta\sin\theta\\\\-\sin\theta\cos\theta&\cos^2\theta\end{bmatrix}+[sin⁡2θ−sin⁡θcos⁡θsin⁡θcos⁡θsin⁡2θ]\begin{bmatrix}\sin^2\theta&-\sin\theta\cos\theta\\\\\sin\theta\cos\theta&\sin^2\theta\end{bmatrix}

=[cos⁡2θ+sin⁡2θcos⁡θsin⁡θ−sin⁡θcos⁡θ−sin⁡θcos⁡θ+sin⁡θcos⁡θcos⁡2θ+sin⁡2θ]\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}

=[10\01]\begin{bmatrix}1&0\\\0&1\end{bmatrix}