Let $d \in \mathbb{R}$,and $A = \begin{bmatrix} -2 & 4+d & \sin \theta - 2 \\ 1 & \sin \theta + 2 & d \\ 5 & 2\sin \theta - d & -\sin \theta + 2 + 2d \end{bmatrix}$,where $\theta \in [0, 2\pi]$. If the minimum value of $\det(A)$ is $8$,then a value of $d$ is

Let $A$ be a matrix of order $2 \times 2$,whose entries are from the set $\{0, 1, 2, 3, 4, 5\}$. If the sum of all the entries of $A$ is a prime number $p$,where $2 < p < 8$,then the number of such matrices $A$ is:

For $M=\left[\begin{array}{ll}3 & -4 \\ 1 & -1\end{array}\right]$ and for any $n \in N$,the matrix $M^{n+1}-M^n=$

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix}$ where each of $a, b$,and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2 & -1 \\ 3 & 0 & k \end{bmatrix}$ and $f(x) = x^3 - 2x^2 - \alpha x + \beta = 0$. If $A$ satisfies $f(A) = 0$,then:

If $A$ and $B$ are two square matrices of order $3$ such that $AB = A$ and $BA = B$,and matrices $X$ and $Y$ are defined as $X = A^4 + B^4$ and $Y = A^{10} + B^{10}$,then the matrix $X - Y$ is: