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 + A^T = I

Answer 1

\[Given: A = \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix}\] 

\[ A^T = \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix}\] 

\[Now, \] 

\[A + A^T = I\] 

\[ \Rightarrow \begin{bmatrix}\cos x & \sin x \\ - \sin x & \cos x\end{bmatrix} + \begin{bmatrix}\cos x & - \sin x \\ \sin x & \cos x\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] 

\[ \Rightarrow \begin{bmatrix}\cos x + \cos x & \sin x - \sin x \\ - \sin x + \sin x & \cos x + \cos x\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] 

\[ \Rightarrow \begin{bmatrix}2\cos x & 0 \\ 0 & 2\cos x\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\] 

\[ \Rightarrow 2\cos x = 1 \] 

\[ \Rightarrow \cos x = \frac{1}{2}\] 

\[ \Rightarrow x = \frac{\pi}{3}\]