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

$M^2$ $-N^2$ $-M^2$ $MN$

$-M^2$

We know $M^T=-M$ $N^T=-N$ $MN = NM$ So $M^{-1}N^{-1}=N^{-1}M^{-1}$ $M^2 N^2 (M^TN)^{-1} (MN^{-1})^T$ $=M^2 N^2 (N^{-1} ×(M^T)^{-1})×((N^{-1})^T M^T)$ $=-M^2N^2N^{-1}M^{-1}N^{-1}M$ $=-M^2$