From the following,find the correct relation.

What must be the matrix $X$ if $2X + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 8 \\ 7 & 2 \end{bmatrix}$?

Assume $X, Y, Z, W$ and $P$ are matrices of order $2 \times n, 3 \times k, 2 \times p, n \times 3$ and $p \times k$ respectively. If $n=p,$ then the order of the matrix $7X - 5Z$ is

If $P = \begin{bmatrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2} \end{bmatrix}$,$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$ and $Q = PAP^T$,then find $P^T Q^{2015} P$.

Let $A = [a_{ij}]$ be a square matrix of order $3$ such that $a_{ij} = 2^{j-i}$,for all $i, j = 1, 2, 3$. Then,the matrix $A^{2} + A^{3} + \ldots + A^{10}$ is equal to

If $P = \begin{bmatrix} \cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4} \end{bmatrix}$ and $X = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}$, then $P^3 X$ is equal to: