Let $A$ be a non-zero periodic matrix with period $4$ and $A^{12} + B = I$,where $I$ is the identity matrix and $B$ is any square matrix of the same order as $A$. The matrix product $AB$ is equal to:

If $A$ is a square matrix of order $3$ with $|A| = 2$,then the value of $|(A - A^T)^5| + |(A^T - A)^3|$ is-

Let $M = \begin{bmatrix} 0 & -\alpha \\ \alpha & 0 \end{bmatrix}$,where $\alpha$ is a non-zero real number and $N = \sum_{k=1}^{49} M^{2k}$. If $(I - M^2)N = -2I$,then the positive integral value of $\alpha$ is

If $C$ and $D$ are two $n \times n$ non-singular matrices over the set of real numbers $\mathbb{R}$ such that $CD = -DC$,then $n$ is:

For any $3 \times 3$ matrix $M$,let $| M |$ denote the determinant of $M$. Let $E=\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{bmatrix}$,$P=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ and $F=\begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix}$. If $Q$ is a nonsingular matrix of order $3 \times 3$,then which of the following statements is (are) $TRUE$?
$(A)$ $F = PEP$ and $P^2 = I$
$(B)$ $| EQ + PFQ^{-1} | = | EQ | + | PFQ^{-1} |$
$(C)$ $|(EF)^3| > |EF|^2$
$(D)$ The sum of the diagonal entries of $P^{-1}EP + F$ is equal to the sum of the diagonal entries of $E + P^{-1}FP$

The value of $\left|\begin{array}{cc}\log _5 729 & \log _3 5 \\ \log _5 27 & \log _9 25\end{array}\right| \times \left|\begin{array}{cc}\log _3 5 & \log _{27} 5 \\ \log _5 9 & \log _5 9\end{array}\right|$ is