In case of option (A),

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

Now, $${N^T}{M^T}N = \left\{ \matrix{ {N^T}MN,\,when\,{M^T} = M \hfill \cr - {N^T}MN,\,when\,{M^T} = - M \hfill \cr} \right.$$

$$\therefore$$ N^TMN is symmetric or skew symmetric according as M is symmetric or skew symmetric.

In case of option (B),

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

$$ = {N^T}{M^T} - {M^T}{N^T}$$

$$ = NM - MN$$ [$$\because$$ $${M^T} = M,\,{N^T} = N$$]

$$ = - (MN - NM)$$

$$\therefore$$ $$MN - NM$$ is skew symmetric matrix.

In case of option (C),

$${(MN)^T} = {N^T}{M^T} = NM$$ [$$\because$$ $${M^T} = M,\,{N^T} = N$$]

Now, MN cannot always be equal to NM

$$\therefore$$ MN is not symmetric matrix.

In case of option (D),

$$(Adj\,M)(Adj\,N) = Adj(NM) \ne Adj(MN)$$

Therefore, (C) and (D) are the correct options.