Find x such that A + A^T = I

Given:

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

Step 1: Find A^T

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

Step 2: Compute A + A^T

\[ A + A^T = \begin{bmatrix} 2\cos x & 0 \\ 0 & 2\cos x \end{bmatrix} \]

Step 3: Equate with Identity Matrix

\[ A + A^T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \]

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

Step 4: Find x

\[ x = \frac{\pi}{3} \quad \text{(since } 0 < x < \frac{\pi}{2}) \]

Final Answer:

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