Let $A$ be a square matrix of order $2$ such that $|A|=2$ and the sum of its diagonal elements is $-3$ . If the points $(x, y)$ satisfying $A^2+x A+y I=0$ lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is e and the length of the latus rectum is $\ell$, then $\mathrm{e}^4+\ell^4$ is equal to……………………… 

If $A$ is a symmetric matrix and $B$ is a skew-symmetrix matrix such that $A + B = \left[ {\begin{array}{*{20}{c}}
2&3\\
5&{ – 1}
\end{array}} \right]$ , then $AB$ is equal to

Show that the matrix $B ^{\prime}A B$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.

If $D = \left| {\begin{array}{*{20}{c}}   {{a_1}}&{{b_1}}&{{c_1}} \\    {{a_2}}&{{b_2}}&{{c_2}} \\    {{a_3}}&{{b_3}}&{{c_3}}  \end{array}} \right|$ and  $D' = \left| {\begin{array}{*{20}{c}}   {{a_1} + p{b_1}}&{{b_1} + q{c_1}}&{{c_1} + r{a_1}} \\    {{a_2} + p{b_2}}&{{b_2} + q{c_2}}&{{c_2} + r{a_2}} \\    {{a_3} + p{b_3}}&{{b_3} + q{c_3}}&{{c_3} + r{a_3}}  \end{array}} \right|$ then

The product of a matrix and its transpose is an identity matrix. the value of determinant of this matrix is