If $A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$ and $\alpha, \beta \in \mathbb{R}$ are such that $\alpha A^2 - \beta A = 2I$,then $\alpha^2 + \beta =$

If $P = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 3 & 4 & 5 \end{bmatrix} \begin{bmatrix} -1 & -2 \\ -2 & 0 \\ 0 & -4 \end{bmatrix} \begin{bmatrix} -4 & -5 & -6 \\ 0 & 0 & 1 \end{bmatrix}$,then $P_{22} = $

If the matrix $AB = O$,then

If $A = \begin{bmatrix} 3 & \sqrt{3} & 2 \\ 4 & 2 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 2 & -1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$,verify that $(kB)^{\prime} = kB^{\prime}$,where $k$ is any constant.

If $A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}$,then the value of $\alpha$ for which $A^2 = B$ is:

If $A$ and $B$ are skew-symmetric matrices of the same order,then $AB - BA$ is a . . . . . . .