Mathematics - Matrices Question with Solution | TestHub

Given, $M^T=-M,N^T=-N$ and $MN=NM$ $$\begin{array}{rl}\therefore & M^2{N}^2{\left(M^{T}N\right)}^{-1}{\left(M{N}^{-1}\right)}^T\\ & =M^2{N}^2{N}^{-1}{\left(M^T\right)}^{-1}{\left(N^{-1}\right)}^T\cdot M^T\\ & =M^{2}N\left(N{N}^{-1}\right)(-M{)}^{-1}{\left(N^T\right)}^{-1}(-M)\\ & =M^{2}N\left(-M^{-1}\right)(-N{)}^{-1}(-M)\\ & =-M^{2}N{M}^{-1}{N}^{-1}M\\ & =-M\cdot (MN)M^{-1}{N}^{-1}M\\ & =-M(NM)M^{-1}{N}^{-1}M\\ & =-MN\left(N{M}^{-1}\right)N^{-1}M\\ & =-M\left(N{N}^{-1}\right)M=-M^2\end{array}$$ Note Here, non-singular word should not be used, since there is no non-singular $3\times 3$ skew-symmetric matrix.