Let `Ma n dN` be two `3xx3` non singular skew-symmetric matrices such that `M N=N M` If `P^T` denote the transpose of `P`, then `M^2N^2(M^TN)^(-1)(M N^(-1))^T` is equal to
a. `M^2`
b. `-N^2`
c. `-M^2`
d. `M N`

Options

A. `M^(2)`
B. `-N^(2)`
C. `-M^(2)`
D. `MN`

Answer

C

Explanation

Given `MN=NM`
`M^(2)N^(2) (M^(T)N)^(-1) (MN^(-1))^(T)`
`=M^(2)N^(2) N^(-1) (M^(T))^(-1) (N^(-1))^(T). M^(T)`
`=M^(2)N. (M^(T))^(-1) (N^(-1))^(T) M^(T)`
`=-M^(2)N(M)^(-1) (N^(T))^(-1) M^(T)`
`=+ M^(2)NM^(-1) N^(-1) M^(T)`
`=-MNM M^(-1) N^(-1) M" "( :' MN =NM)`
`=-MN N^(-1) M=-M^(2)`