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 symmetricor 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 of skew-symmetric
B. `MN-NM` is skew-symmetric 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 invertible matrices M and N

Answer

C, D

Explanation

(a) `(N^(T) MN)^(T) = N^(T) M^(T) (N^(T))^(T)=N^(T)M^(T)N=N^(T) MN`
or `-N^(T) MN` According as M is symmetric ro
skew-symmetric.
`therefore` Correct.
(b) `(MN-NM)^(T) = (MN)^(T)- (NM)^(T) =N^(T) M^(T) - M^(T)N^(T) `
`= NM=MN ` [`because`+ M, N are symmetric]
`=-(MN-NM)`
`therefore ` correct
(c) `(MN)^(T) = N^(T) M^(T) = NMneMN` [`because` M, N are symmetric]
`therefore` Incorrect.
(d) `(adj M) (adjN) = adj (NM) ne adj(MN)`
`therefore ` Incorrect.