For `3xx3` matrices `M \ a n d \ N ,` which of the following statement (s) is (are) NOT correct ?
Statement - I : `N^T M N` is symmetric or skew-symmetric, according as `M` is symmetric or skew-symmetric.
Statement - II : `M N-N M` is skew-symmetric for all symmetric matrices `Ma n dN`.
Statement - III : `M N` is symmetric for all symmetric matrices `M a n dN`.
Statement - IV : `(a d jM)(a d jN)=a d j(M N)` for all invertible matrices `Ma n dN`.

Options

A. `N^(T)MN` is symmetric or skew-symmetric, according as M is symmetric or skew-symmetric
B. `MN-NM` is skew0symmetric for all symmetric matrices M and N
C. MN is symmetric for all symmetric matrices M and N
D. (adj M ) (adj N) = adj (MN) for all inveriblr matrices M and N.

Answer

C, D

Explanation

(1) `(N^(T)MN)^(T)=N^(T)M^(T)N=N^(T)MN` if `M` is symmetric and is `-N^(T)MN` if `M` is skew symmetric
(2) `(MN-NM)^(T)=N^(T) M^(T)-M^(T)N^(T)=NM-MN=-(MN-NM)`.
So, `(MN-NM)` is skew symmetric
(3) `(MN)^(T)=N^(T)M^(T)=NM ne MN` if `M` and `N` are symmetric. So, MN is not symmetric
(4) (adj. M) (adj. N) = adj `(NM) ne` adj `(MN)`