Question

If $A=\left[\begin{array}{cc}\cos x& -\sin x\\ \sin x& \cos x\end{array}\right]$, find AA^T

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}A=\left[\begin{array}{cc}\cos x& -\sin x\\ \sin x& \cos x\end{array}\right] \\ \Rightarrow A^T=\left[\begin{array}{cc}\cos x& \sin x\\ -\sin x& \cos x\end{array}\right] \\ A{A}^T=\left[\begin{array}{cc}\cos x& -\sin x\\ \sin x& \cos x\end{array}\right]\left[\begin{array}{cc}\cos x& \sin x\\ -\sin x& \cos x\end{array}\right] \\ \Rightarrow A{A}^T=\left[\begin{array}{cc}{\cos }^{2}x+{\sin }^{2}x& \cos x\sin x-\sin x\cos x\\ \cos x\sin x-\sin x\cos x& {\sin }^{2}x+{\cos }^{2}x\end{array}\right] \\ \Rightarrow A{A}^T=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \end{gathered}$$