Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix whose elements are given by \( a_{ij} = \frac{(i + 2j)^2}{3} \). Then \( A^{\prime} \) is given by :
If \( A = \begin{bmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{bmatrix} \) and \( A + A^{\prime} = I \), then the value of \( x \in \left[ 0, \dfrac{\pi}{2} \right] \) is