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Question
If \[A = \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix}\] , find x satisfying 0 < x < \[\frac{\pi}{2}\] when A + AT = I
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Solution
\[Given: A = \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix}\]
\[ A^T = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\]
\[Now, \]
\[A + A^T = I\]
\[ \Rightarrow \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix} + \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}\cos x + \cos x & \sin x - \sin x \\ - \sin x + \sin x & \cos x + \cos x\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}2\cos x & 0 \\ 0 & 2\cos x\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]
\[ \Rightarrow 2\cos x = 1 \]
\[ \Rightarrow \cos x = \frac{1}{2}\]
\[ \Rightarrow x = \frac{\pi}{3}\]
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