If the matrix equation $\begin{bmatrix} x+y & -2 \\ 7+z & x-y \end{bmatrix} = \begin{bmatrix} -7 & -2 \\ 5 & 0 \end{bmatrix}$ holds,then find the value of $2x + 4y + 2z$.

If $\begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix}$,then $\frac{x^2+y^2+z^2}{\gamma} =$

Let $P = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 9 & 3 & 1 \end{bmatrix}$ and $Q = [q_{ij}]$ be two $3 \times 3$ matrices such that $Q - P^5 = I_3$. Then $\frac{q_{21} + q_{31}}{q_{32}}$ is equal to

If $A$ is a square matrix such that $A^2 = A$,then $(I + A)^3 - 8A =$ . . . . . . .

If $A = \begin{bmatrix} 5 & -3 \\ 2 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 6 & -4 \\ 3 & 6 \end{bmatrix}$,then $A - B = $

If $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$,then $A^{100} = $