Let $A$ be a $3 \times 3$ matrix of non-negative real elements such that $A\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=3\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$. Then the maximum value of $\operatorname{det}(\mathrm{A})$ is……………………

If $A = \left( {\begin{array}{*{20}{c}}2&{ – 1}\\{ – 1}&2\end{array}} \right)$ and $I$ is the unit matrix of order $2$, then ${A^2}$ equals

A trust fund has Rs. $30,000$ that must be invested in two different types of bonds. The first bond pays $5 \%$ interest per year, and the second bond pays $7 \%$ interest per year. Using matrix multiplication, determine how to divide Rs. $30,000$ among the two types of bonds. If the trust fund must obtain an annual total interest of  Rs. $2000$.

Let $A = \left[ {\begin{array}{*{20}{c}}
p&{13}\\
{ – 13}&p
\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{c}}
{4q}&{85}\\
{ – 2}&1
\end{array}} \right]$  where  $p,q \in N$. It is given that $\left| A \right| = \left| B \right|$ and  $p,q \in[1,1000]$. Then total number of ordered pairs $(p,q)$ is

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

If the unit costs of the above three commodities are  $\mathrm{Rs} $. $2.00, $ $\mathrm{Rs} $.  $1.00$ and $50$ paise respectively. Find the gross profit.