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

I. $N^T MN$ is symmetric skew-symmetric, according as M is symmetric or skew-symmetric.

II. $MN- NM$ is skew symmetric for all symmetric matrices M and N.

III. MN is symmetric for all symmetric matrices M and N.

IV. (adj M) (adj N) = adj (MN) for all invertible matrices M & N.

III & IV

Clearly, $(N^TMN)^T = N^TM^T (N^T)^T = N^TM^TN$

$⇒(N^TMN)^T=\left\{\begin{matrix}N^T MN,\, if\, M^T = M\, i.e.\, M\, is\, symmetric\\-N^T MN,\, if\, M^T =-M\, i.e.\, M\, is\, skew-symmetric\end{matrix}\right.$

So, statement I is correct.

$(MN-NM)^T=(MN)^T -(NM)^T$

$=N^TM^T – M^TN^T$

$=NM-MN$   [∵ N & M are symmetric matrices]

$=-(MN-NM)$

∴ MN - NM is skew-symmetric matrix.

If M and N are symmetric matrices, then 

$(MN)^T =N^TM^T = NM≠ MN$

So, MN is not a symmetric matrix.

Hence, statement III is not correct.

We know that adj (MN) = (adj N) (adj M). So, statement IV is not correct.