Question
If $f(\alpha)=A=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]$, find
if(-α) + f(α)
if(-α) + f(α)
$\begin{aligned} & =\left[\begin{array}{ccc}\cos \alpha+\cos \alpha & \sin \alpha-\sin \alpha & 0+0 \\ -\sin \alpha+\sin \alpha & \cos \alpha+\cos \alpha & 0+0 \\ 0+0 & 0+0 & 1+1\end{array}\right] \\ & =\left[\begin{array}{ccc}2 \cos \alpha & 0 & 0 \\ 0 & 2 \cos \alpha & 0 \\ 0 & 0 & 2\end{array}\right]\end{aligned}$
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$\begin{aligned} & {\left[\begin{array}{lll}0 & -1 & 4\end{array}\right]\left\{2\left[\begin{array}{cc}4 & 5 \\ 3 & 6 \\ 2 & -1\end{array}\right]+3\left[\begin{array}{cc}4 & 3 \\ 1 & 4 \\ 0 & -1\end{array}\right]\right\}} \\ & =\left[\begin{array}{ll}x & y\end{array}\right] .\end{aligned}$
$\left[\begin{array}{ccc}0 & 4 & 7 \\ -4 & 0 & -3 \\ -7 & 3 & 0\end{array}\right]$