Question

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

• A. $N^{T}MN$ is symmetric or skew-symmetric, according as $M$ is symmetric or skew-symmetric
• B. $MN-NM$ is symmetric for all symmetric matrices $M$ and $N$
• C. $MN$ is symmetric for all symmetric matrices $M$ and $N$
• D. $(adjM)(adjN)$ = $adj(MN)$ for all invertible matrices $M$ and $N$

Answer 1

Answer: (D)

(a) ${\left(N^{T}MN\right)}^T=N^T{M}^T{\left(N^T\right)}^T=N^T{M}^{T}N$, is symmetric if
$M$ 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) $(\mathrm{adj}MN)=(\mathrm{adj}N)\cdot (\mathrm{adj}M)$
$\therefore$ Not correct.