Let $M = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}$ and $I$ be the identity matrix of order $3$. Then $M^2 - 4M =$

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 $P^T Q^{2015} P$.

Consider the following information regarding the number of men and women workers in three factories $I, II$ and $III$.

Factory	Men and Women Workers
$I$	$30$ Men,$25$ Women
$II$	$25$ Men,$31$ Women
$III$	$27$ Men,$26$ Women

Represent the above information in the form of a $3 \times 2$ matrix. What does the entry in the third row and second column represent?

Let $\alpha, \beta, \gamma$ be real numbers. If $\begin{bmatrix} 7 & 5 & \alpha \\ \beta & 2 & 11 \\ 3 & \gamma & 1 \end{bmatrix} \begin{bmatrix} 1 \\ 3 \\ 2 \end{bmatrix} = \begin{bmatrix} \alpha+\beta \\ -2\alpha+\beta-2\gamma \\ \alpha+2\beta+3\gamma \end{bmatrix}$, then find the value of $100+\frac{2\alpha+11\beta}{\gamma}$.

The matrix $A = \begin{bmatrix} 1 & -3 & -4 \\ -1 & 3 & 4 \\ 1 & -3 & -4 \end{bmatrix}$ is nilpotent of index:

If $A = \begin{bmatrix} 4 & 2 \\ -1 & 1 \end{bmatrix}$ and $I$ is the identity matrix of order $2$,then $(A - 2I)(A - 3I) = $