Question

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

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

Answer 1

Answer: (C)

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