If $P = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$, then $P^5$ is equal to

If $P = \begin{bmatrix} 1 & 2 & 1 \\ 1 & 3 & 1 \end{bmatrix}$ and $Q = P P^{T}$, then the value of the determinant of $Q$ is:

Let $A$ and $B$ be two symmetric matrices of the same order. Then, the matrix $AB - BA$ is

If $A$ is a nilpotent matrix of index $2$,then $A(I_2+A)^{51}$ is equal to (where $I_2$ is the identity matrix of order $2$)

If $A = \begin{bmatrix} 0 & 2 \\ 3 & -4 \end{bmatrix}$ and $hA = \begin{bmatrix} 0 & 3a \\ 2b & 24 \end{bmatrix}$, then the values of $h, a, b$ are respectively

$A=\left[\begin{array}{ccc}a^2 & 15 & 31 \\ 12 & b^2 & 41 \\ 35 & 61 & c^2\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 a & 3 & 5 \\ 2 & 2 b & 8 \\ 1 & 4 & 2 c-3\end{array}\right]$ are two matrices such that the sum of the principal diagonal elements of both $A$ and $B$ are equal,then the product of the principal diagonal elements of $B$ is