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If $\mathrm{A}=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$ then what is $\mathrm{AA}^{\mathrm{T}}$ equal to (where $\begin{array}{ll} \left.\mathrm{A}^{\mathrm{T}} \text { is the transpose of } \mathrm{A}\right) ?\end{array}$
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Identify matrix
$A=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$
$A A^{T}=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$
$=\left[\begin{array}{cc}\cos ^{2} \alpha+\sin ^{2} \alpha & -\cos \alpha \sin \alpha+\sin \alpha \cos \alpha \\ -\sin \alpha \cos \alpha+\sin \alpha \cos \alpha & \sin ^{2} \alpha+\cos ^{2} \alpha\end{array}\right]$
$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
$A A^{T}=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$
$=\left[\begin{array}{cc}\cos ^{2} \alpha+\sin ^{2} \alpha & -\cos \alpha \sin \alpha+\sin \alpha \cos \alpha \\ -\sin \alpha \cos \alpha+\sin \alpha \cos \alpha & \sin ^{2} \alpha+\cos ^{2} \alpha\end{array}\right]$
$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
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