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For 3x3 matrices M and N , which of the following statement(s) is (are) not correct?

$N^{T}MN$ is symmetric or skew-symmetric , according as M is symmetric or skew-symmetric

B) MN-NM is symmetric for all symmetric matrices M andN

C) M N is symmetric for all symmetric matrices M and N

D) (adj M) (adj N) = adj(MN) for all invertible matrices M and N

Option C,D

(a) $(N^{T}MN )^{T}=N^{T}M^{T}(N^{T})^{T}$

= $N^{T}M^{T}N$ , is symmetric is M is symmetric and skew symmetric is M is skew-symmetric

(b) $(MN-NM)^{T}=(MN)^{T}-(NM)^{T}$

=NM-MN=-(MN-NM)

$\therefore$ skew-symmetric , when M and N are symmetric

$\therefore$ not correct

(d) (adj MN)= (adj N)-(adj M)

$\therefore$ not correct

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