Question

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

• A. $N^TMN$ is symmetric or skew-symmetric, according 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: (D)

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

Answer 2

(a) $(N^TMN)^T =N^TM^T(N^T)^T =N^TM^TN,$ is symmetric if
M is symmetric and skew-symmetric, if M is skew-symmetric.
(b) (MN - N M)$^T$ = (.M N )$^T$ - (NM)$^T$
\hspace20mm = NM - MN = - (MN - NM)
$\therefore \, \,$ Skew-symmetric, when M and N are symmetric
(c)$(MN)^T=N^TM^T=NM \ne MN$
$\therefore$ , , Not correct
(d) (adj MN) =(adj N).(adj M)
$\therefore \, \, \, \, \,$. Not correct.