Let $A = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}$ and $B = \begin{bmatrix} 9^2 & -10^2 & 11^2 \\ 12^2 & 13^2 & -14^2 \\ -15^2 & 16^2 & 17^2 \end{bmatrix}$,then the value of $A^{\prime} BA$ is.

If $A = \begin{bmatrix} 0 & i \\ -i & 0 \end{bmatrix}$,then the value of $A^{40}$ is

If $AB = C$,then the dimensions of matrices $A, B, C$ are:

If $A = \begin{bmatrix} 0 & -\tan(\frac{\theta}{2}) \\ \tan(\frac{\theta}{2}) & 0 \end{bmatrix}$ and $(I_{2} + A)(I_{2} - A)^{-1} = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$,then $13(a^{2} + b^{2})$ is equal to ...........

Let $A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}$,$B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}$,and $C = \begin{bmatrix} -2 & 5 \\ 3 & 4 \end{bmatrix}$. Find $A + B$.

If $A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$,then prove that $A^n = \begin{bmatrix} 1+2n & -4n \\ n & 1-2n \end{bmatrix}$,where $n$ is any positive integer.