Given expression : M^{2} N^{2} (M^{T}N) ^{-1} (MN^{-1})^{T}
^{ }= M^{2}N^{2 }(-MN)^{-1 }(MN^{-1})^{T} [ Since M is a skew symmetric matrix , so M^{T }= -M]
= - M^{2}N^{2 } . N^{-1}M^{-1} .(N^{T})^{-1} M^{T}
^{ } [Since for any two matrices A and B , (AB)^{-1} = B^{-1}A^{-1} and
(AB)^{T }= B^{T} A^{T} and
(A^{-1})^{T }= (A^{T})^{-1} ]
= - M^{2} N^{2} N^{-1} M^{-1} (-N)^{-1}(-M)
= - M. M N M^{-1 }(-N)^{-1}(-M)^{ }[ Since for any matrix A , A A^{-1 }= I where I is identity matrix]
= - M N M M^{-1 }(-N)^{-1}(-M) [Since given in the question MN = NM]
^{ }= - M^{2 }[Since middle M,M^{-1 }and N,N^{-1} are eliminated using A A^{-1 }= I]
Hence C) option should be the correct option.