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

Let $A=\left(\begin{array}{rrr}1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1\end{array}\right)$ and $B=7 A^{20}-20 A^{7}+2 I$, where $I$ is an identity matrix of order $3 \times 3$ If $B=\left[b_{i j}\right]$, then $b_{13}$ is equal to $….$

Construct a $2 \times 2$ matrix, $A=\left[a_{ij}\right]$, whose elements are given by : $a_{i j}=\frac{(i+j)^{2}}{2}$

If $A=\left[\begin{array}{rrr}0 & 6 & 7 \\ -6 & 0 & 8 \\ 7 & -8 & 0\end{array}\right], B=\left[\begin{array}{lll}0 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 0\end{array}\right], C=\left[\begin{array}{c}2 \\ -2 \\ 3\end{array}\right]$

Calculate $AC$, $BC$ and $(A + B)C$. Also, verify that $(A+B)\,C=A C+B C$

The order of $[x\,y\,z]\,\,\left[ {\begin{array}{*{20}{c}}a&h&g\\h&b&f\\g&f&c\end{array}} \right]\,\left[ \begin{array}{l}x\\y\\z\end{array} \right]$ is