Question

For $3\times 3$ matrices M and N, which of the following sttements (s) is/are not correct?

• A. $N^T$ MN is symmetric or skew – symmetric, accroding as M is symmetric or skew – symmetric.
• B. MN – NM is 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 1

Answer: (B), (C), (D)

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

(a) $(N^{T}MN{)}^T=N^T{M}^T(N^T{)}^T=N^T{M}^{T}N,$ is symmtric is M is symmetric and skew – symmetric, if 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.
(c) $(MN{)}^T=N^T{M}^T=NM\ne MN$
$\therefore$ Not correct
(d) (adj MN) = (adj N). (adj M)
$\therefore$ Not correct.