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

$(a)\;M^2\qquad(b)\;-N^2\qquad(c)\;-M^2\qquad(d)\;MN$

As a skew symmetric matrix of order 3 cannot be non singular,therefore the data given in the question is inconsistent.
We have
$M^2N^2(MTN)^{-1}(MN^{-1})^T=M^2N^2N^{-1}(MT)^{-1}(N^{-1})^TM^T$
$\Rightarrow M^2N(M^T)^{-1}(N^{-1})^TM^T=-M^2NM^{-1}N^{-1}M$
$M^T=-M,N^T=-N)$ and $(N^{-1})^T=(NT)^{-1}$
$MN=NM$
$\Rightarrow -M(NM)(NM)^{-1}M$
$\Rightarrow -MM$
$\Rightarrow -M^2$
Hence (c) is the correct answer.