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

Let $\left| \begin{array}{ccc} (a-x)^2 & (a-y)^2 & (a-z)^2 \\ (b-x)^2 & (b-y)^2 & (b-z)^2 \\ (c-x)^2 & (c-y)^2 & (c-z)^2 \end{array} \right| = \frac{-351}{8}$. If $x, y, z$ are the roots of the equation $8t^3 - 62t^2 + 43t - 7 = 0$ and $a, b, c$ are distinct numbers,then the value of $|(a-b)(b-c)(c-a)|$ is:

Let $S = \{ m \in \mathbb{Z} : A^{m^2} + A^m = 3I - A^{-6} \}$,where $A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. Then $n(S)$ is equal to

For $A = \begin{bmatrix} 0 & 0 & -2 \\ 0 & -2 & 0 \\ -2 & 0 & 0 \end{bmatrix}$,which of the following is true?

Let $\alpha$ be a root of the equation $x^{2}+x+1=0$ and the matrix $A=\frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & \alpha^{2} \\ 1 & \alpha^{2} & \alpha^{4} \end{bmatrix}$,then the matrix $A^{31}$ is equal to

Let $A, B$ be two $3 \times 3$ matrices and $C$ be a $3 \times 3$ identity matrix such that $AB-C$ is a non-singular matrix. Let $D=(AB-C)^{-1}$. Then, consider the following statements.
Statement $I$: $\operatorname{det}(BA)=\operatorname{det}(BA-C) \operatorname{det}(BDA)$
Statement $II$: $ABD=DAB$
Which of the above statements is (are) true?