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 $M^2{N}^2{\left(M^{T}N\right)}^{-1}{\left(M{N}^{-1}\right)}^T$ is equal to

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

Answer 1

Answer: (C)

$-M^2$

$M^2{N}^2\left(M^{T}N{)}^{-1}\right(M{N}^{-1}{)}^T=M^2{N}^2{N}^{-1}\left(M^T{)}^{-1}\right(N^{-1}{)}^T{M}^T$

$\Rightarrow -M^2{N}^2{N}^{-1}{M}^{-1}{N}^{-1}M$ ....Given M and N are skew-symmetric matix

$\Rightarrow -M^{2}N{M}^{-1}{N}^{-1}M$

$\Rightarrow -M^{2}N{N}^{-1}{M}^{-1}M$ (using $M^{-1}{N}^{-1}=N^{-1}{M}^{-1}$)

$\Rightarrow -M^{2}II$

$\therefore M^2{N}^2\left(M^{T}N{)}^{-1}\right(M{N}^{-1}{)}^T=-M^2$