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

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

Answer 1

The correct option is C $-M^2$

Given $M$ and $N$ are skew-symmetric matrices
$\Rightarrow M^T=-M,N^T=-N$
Also, $(MN{)}^2=(MN\left)\right(MN)=M(NM)N=M(MN)N=M^2{N}^2$ ...... $[\because MN=NM]$
Now,

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