Let $A = \begin{bmatrix} x & 1 \\ 1 & 0 \end{bmatrix}$,$x \in \mathbb{R}$ and $A^{4} = [a_{ij}]$. If $a_{11} = 109$,then $a_{22}$ is equal to

If $I$ is the identity matrix of order $2$ and $A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$, then for $n \geq 1$, mathematical induction gives:

$A = [a_{ij}]_{m \times n}$ is a square matrix,if

Let $Q = \begin{bmatrix} \cos \frac{\pi}{4} & -\sin \frac{\pi}{4} \\ \sin \frac{\pi}{4} & \cos \frac{\pi}{4} \end{bmatrix}$ and $x = \begin{bmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{bmatrix}$. Then $Q^{3} x$ is equal to:

Compute the indicated product: $\left[\begin{array}{cc}2 & 1 \\ 3 & 2 \\ -1 & 1\end{array}\right] \times \left[\begin{array}{ccc}1 & 0 & 1 \\ -1 & 2 & 1\end{array}\right]$

If $A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{bmatrix}$ is a matrix satisfying the equation $A A^T = 9 I$,where $I$ is the identity matrix,then $a^2 + b^2 =$