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)

(a) $\left(N^{T} M N\right)^{T}=N^{T} M^{T}\left(N^{T}\right)^{T}=N^{T} M^{T} N$, is symmetric if
$M$ is symmetric and skew-symmetric, if $M$ is skew-symmetric.
(b) $(M N-N M)^{T} =(M N)^{T}-(N M)^{T} $
$=N M-M N=-(M N-N M)$
$\therefore$ Skew-symmetric, when $M$ and $N$ are symmetric.
(c) $(M N)^{T}=N^{T} M^{T}=N M \neq M N$
$\therefore$ Not correct.
(d) $(\operatorname{adj} M N)=(\operatorname{adj} N) \cdot(\operatorname{adj} M)$
$\therefore$ Not correct.