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

If unit sale prices of $x, \,y$ and $z$ are Rs. $2.50$, Rs. $1.50$ and Rs. $1.00,$ respectively, find the total revenue in each market with the help of matrix algebra.

The set of all $2 \times 2$ matrices over the real numbers is not a group under matrix multiplication because

Let $\mathrm{p}$ be an odd prime number and $\mathrm{T}_{\mathrm{p}}$ be the following set of $2 \times 2$ matrices :

$T_p=\left\{A=\left[\begin{array}{ll}\mathrm{a} & \mathrm{b} \\ \mathrm{c} & \mathrm{a}\end{array}\right]: \mathrm{a}, \mathrm{b}, \mathrm{c} \in\{0,1, \ldots ., \mathrm{p}-1\}\right\}$

$1.$ The number of $A$ in $T_p$ such that $A$ is either symmetric or skew-symmetric or both, and $\operatorname{det}(\mathrm{A})$ divisible by $\mathrm{p}$ is

$(A)$ $(\mathrm{p}-1)^2$  $(B)$ $2(\mathrm{p}-1)$

$(C)$ $(\mathrm{p}-1)^2+1$  $(D)$ $2 \mathrm{p}-1$

$2.$ The number of $A$ in $T_p$ such that the trace of $A$ is not divisible by $p$ but det $(A)$ is divisible by $p$ is [Note: The trace of a matrix is the sum of its diagonal entries.]

$(A)$ $(\mathrm{p}-1)\left(\mathrm{p}^2-\mathrm{p}+1\right)$ $(B)$ $\mathrm{p}^3-(\mathrm{p}-1)^2$

$(C)$ $(\mathrm{p}-1)^2$ $(D)$ $(p-1)\left(p^2-2\right)$

$3.$ The number of $A$ in $T_p$ such that det $(A)$ is not divisible by $p$ is

$(A)$ $2 \mathrm{p}^2$ $(B)$ $p^3-5 p$ $(C)$ $p^3-3 p$ $(D)$ $p^3-p^2$

Give the answer question $1,2$ and $3.$

For how many value $(s)$ of $x$ in the closed interval $[ – 4,\,\, – 1]$ is the matrix $\left[ {\begin{array}{*{20}{c}}3&{ – 1 + x}&2\\3&{ – 1}&{x + 2}\\{x + 3}&{ – 1}&2\end{array}} \right]$  singular

If $A = \left[ {\begin{array}{*{20}{c}}3&1\\{ – 1}&2\end{array}} \right]$, then ${A^2} = $