If $A$ is a $3 \times 3$ non-zero matrix such that $A^2 = 0$ then the determinant of $(I + A)^{50} - 50A$ is equal to

Suppose $\left|\begin{array}{lll}a & a^{2} & a^{3}-1 \\ b & b^{2} & b^{3}-1 \\ c & c^{2} & c^{3}-1\end{array}\right|=0$, where $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ are distinct real numbers. If ${a}=3$, then the value of $abc$ is:

Suppose $a, b$ and $c$ are integers such that $a>b>c>0$, and $A=\left[\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right]$. Then the value of the determinant of $A$ is

If inverse of the matrix $\left[\begin{array}{cc}2 & -0.5 \\ -1 & x\end{array}\right]$ is $\left[\begin{array}{ll}1 & 1 \\ 2 & 4 \end{array}\right]$, then the value of $x$ is

If $A = \begin{bmatrix} x_1 & x_2 & 7 \\ y_1 & y_2 & y_3 \\ z_1 & 8 & 3 \end{bmatrix}$ is a matrix such that the sum of all three elements along any row, column or diagonal are equal to each other, then the value of determinant of A is:

A $2 \times 2$ matrix is filled with four distinct integers randomly chosen from the set $\{1,2,3,4,5,6\}$. Then the probability that the matrix generated in such a way is singular is

If $A=\left[\begin{array}{ll}1 & 0 \\ \frac{1}{2} & 0\end{array}\right]$ then $A^{2022}$ is