The number of elements in the set $\{A=\begin{bmatrix} a & b \\ 0 & d \end{bmatrix} : a, b, d \in \{-1, 0, 1\} \text{ and } (I-A)^3 = I-A^3 \}$,where $I$ is the $2 \times 2$ identity matrix,is:

$A, P, B$ are $3 \times 3$ matrices. If $|-B|=5, |BA^T|=15, |P^T AP|=-27$, then one of the values of $|P|$ is

Let $\alpha, \beta, \gamma, \delta$ be distinct imaginary roots of $z^5=1$. Find the value of the determinant: $\left| \begin{array}{ccc} e^{\alpha} & e^{2\alpha} & e^{3\alpha+1} \\ e^{\beta} & e^{2\beta} & e^{3\beta+1} \\ e^{\gamma} & e^{2\gamma} & e^{3\gamma+1} \end{array} \right|$.

If $A = \begin{bmatrix} -1 & x & -3 \\ 2 & 4 & z \\ y & 5 & -6 \end{bmatrix}$ is a symmetric matrix and $B = \begin{bmatrix} 0 & 2 & q \\ p & 0 & -4 \\ -3 & r & s \end{bmatrix}$ is a skew-symmetric matrix,then $|A| + |B| - |AB| = $

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 $B$ and $C$ be $n \times n$ matrices such that $A=B+C$, $BC=CB$, and $C^2=0$ (where $0$ is the null matrix). Then, $B^{2020}[B+(2021)C]=$