If $A$ and $B$ are square matrices of the same order such that $AB = A$ and $BA = B$,then $(A + I)^5$ is equal to (where $I$ is the identity matrix).

Let $\omega = - \frac{1}{2} + i\frac{\sqrt{3}}{2}$. Then the value of the determinant $\left| \begin{array}{ccc} 1 & 1 & 1 \\ 1 & -1 - \omega^2 & \omega^2 \\ 1 & \omega^2 & \omega^4 \end{array} \right|$ is

Let $A$ be a $2 \times 2$ real matrix and $I$ be the identity matrix of order $2$. If the roots of the equation $|A-xI|=0$ are $-1$ and $3$,then the sum of the diagonal elements of the matrix $A^2$ is $..............$

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 $A^2+xA+yI=0$ 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 $e^4+\ell^4$ is equal to?

Let $A^{-1} = \begin{bmatrix} 1 & 2017 & 2 \\ 1 & 2017 & 4 \\ 1 & 2018 & 8 \end{bmatrix}$. Then,$|2A| - |2A^{-1}|$ is equal to

Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} \in \{0, 1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then,the variance of $X$ is: