If $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$,then $(aI + bA)^n$ is (where $I$ is the identity matrix of order $2$)

Let $A=\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 3 & 4\end{array}\right]$,$B=\left[\begin{array}{ccc}4 & 0 & -3 \\ -1 & -2 & -3\end{array}\right]$ and $C=\left[\begin{array}{cccc}2 & -3 & 0 & 1 \\ 5 & -1 & -4 & 2 \\ -1 & 0 & 0 & 3\end{array}\right]$,what is $A^T B$ ?

If $A$ is a $3 \times 4$ matrix and $B$ is a matrix such that $A^{\prime}B$ and $BA^{\prime}$ are both defined,then $B$ is of the type:

If $A = \begin{bmatrix} 1 & 2 \\ -3 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 0 \\ 2 & 3 \end{bmatrix}$,then:

If $A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$,then $A^{100} = $ . . . . . . .

$A$ manufacturer produces three products $x, y, z$ which he sells in two markets. Annual sales are indicated below:

Market	$x$	$y$	$z$
$I$	$10,000$	$2,000$	$18,000$
$II$	$6,000$	$20,000$	$8,000$

If the unit sale prices of the above three commodities are $Rs. 2.50, Rs. 1.50$ and $Rs. 1.00$ respectively,and unit costs are $Rs. 2.00, Rs. 1.00$ and $50$ paise respectively,find the total gross profit.