Question

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

• A. $M^2$
• B. $-N^2$
• C. $-M^2$
• D. $MN$

Answer 1

Answer: (C)

Given, $M^T = - M, N^T = - N$ and $M N = N M ....$(i)
$\therefore M^2N^2(M^T N)^{-1} (MN^{-1})^T$
$\Rightarrow M^2N^2N^{-1} (M^T)^{-1}(N^{-1})^T.M^T$
$\Rightarrow M^2N(NN^{-1})(-M)^{-1}(N^T)^{-1}(-M)$
$\Rightarrow M^2N I(-M^{-1})(-N)^{-1} (-M) \, \Rightarrow -M^2NM^{-1}N^{-1}M$
$\Rightarrow -M.(MN)M^{-1}N^{-1}M =-M(NM)M^{-1}N^{-1}M$
$\Rightarrow -MN (NM^{-1})N^{-1}M=- M(NN^{-1})M \Rightarrow -M^2$
NOTE Here, non-singular word should not be used, since there is no non-singular $3 \times 3$ skew-symmetric matrix.