If $A=\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]$, find $A A^{\top}$
Given : $A=\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]$
$\Rightarrow A^{T}=\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]$
$A A^{T}=\left[\begin{array}{cc}\cos x & -\sin x \\ \sin x & \cos x\end{array}\right]\left[\begin{array}{cc}\cos x & \sin x \\ -\sin x & \cos x\end{array}\right]$
$\Rightarrow A A^{T}=\left[\begin{array}{cc}\cos ^{2} x+\sin ^{2} x & \cos x \sin x-\sin x \cos x \\ \cos x \sin x-\sin x \cos x & \sin ^{2} x+\cos ^{2} x\end{array}\right]$
$\Rightarrow A A^{T}=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$
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