If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & -1 \end{bmatrix}$,then ${A^2} = $

If $A = \begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}$,then $A^2 = $

If the matrix $A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 2 & 0 & 2 \end{bmatrix}$,then $A^n = \begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ b & 0 & a \end{bmatrix}$,for $n \in N$,where:

If $A = \begin{bmatrix} 1 & 2 & 3 \\ -2 & 3 & -1 \\ 3 & 1 & 2 \end{bmatrix}$ and $I$ is a unit matrix of $3^{rd}$ order,then $(A^2 + 9I)$ equals

$A$ 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.

If $M = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix}$ and $M^2 - \lambda M - I_2 = 0$,then $\lambda = $