$A$ square matrix $A = [a_{ij}]$ in which $a_{ij} = 0$ for $i \neq j$ and $a_{ij} = k$ (constant) for $i = j$ is called a

Simplify $\cos \theta \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} + \sin \theta \begin{bmatrix} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{bmatrix}$.

If $A = \begin{bmatrix} 0 & 2 & 0 \\ 0 & 0 & 3 \\ -2 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & -4 & 0 \end{bmatrix}$,then the element of the $3^{rd}$ row and $3^{rd}$ column in $AB$ will be:

Find the values of $a, b, c,$ and $d$ from the following equation:
$\begin{bmatrix} 2a+b & a-2b \\ 5c-d & 4c+3d \end{bmatrix} = \begin{bmatrix} 4 & -3 \\ 11 & 24 \end{bmatrix}$

If $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$,show that $A^{2} - 5A + 7I = 0$.

Compute the following: $\left[ {\begin{array}{cc} {{\cos }^2}x & {{\sin }^2}x \\ {{\sin }^2}x & {{\cos }^2}x \end{array}} \right] + \left[ {\begin{array}{cc} {{\sin }^2}x & {{\cos }^2}x \\ {{\cos }^2}x & {{\sin }^2}x \end{array}} \right]$