Question

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

Answer 1

As a skew symmetric matrix of order $3$ cannot be non singular therefore data given is in consistent.

We have,

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

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

$M^T=-M,N^T=-N$ and ${\left(N^{-1}\right)}^T={\left(NT\right)}^{-1}$

$MN=NM$

$\Rightarrow -M\left(NM\right)\left(N{M}^{-1}\right)M$

$\Rightarrow -MN$

$\Rightarrow -M^2$

Hence, the answer is $-M^2.$