Let $M$ and $N$ be two $3 \times 3$ non-singular skew-symmetric matrices such that $MN = NM$. If $P^T$ denotes the transpose of $P$,then $M^2 N^2 (M^T N)^{-1} (M N^{-1})^T$ is equal to

If $A = \begin{bmatrix} \cos^2 \theta & \sin \theta \cos \theta \\ \sin \theta \cos \theta & \sin^2 \theta \end{bmatrix}$,$B = \begin{bmatrix} \cos^2 \phi & \sin \phi \cos \phi \\ \sin \phi \cos \phi & \sin^2 \phi \end{bmatrix}$ and $\theta$ and $\phi$ differ by $\frac{\pi}{2}$,then $AB = $

Let $A$ be a matrix such that $A \cdot \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}$ is a scalar matrix and $|3A| = 108$. Then $A^2$ equals

If $P$ and $Q$ are two non-singular matrices of the same order such that $Q^r = I$,for some integer $r > 1$,then $P^{-1}Q^{r-1}P - P^{-1}Q^{-1}P$ is equal to (where $I$ is the identity matrix and $O$ is the null matrix).

Let $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$. The sum of the prime factors of $|P^{-1}AP - 2I|$ is equal to

If matrix $A = \begin{bmatrix} 1 & 5 \\ 6 & 7 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$,then which of the following statements is incorrect?