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 to as $M$ is symmetric or skew-symmetric
• B. $MN-NM$ is skew-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: (C), (D)

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

$\left(A\right)(N^{T}MN{)}^T=N^T{M}^{T}N=N^{T}MN$, if $M$ is symmetric

$(N^{T}MN{)}^T=N^T{M}^{T}N=-N^{T}MN$, if $M$ is skew symmetric

(B) $(MN-NM{)}^T=N^T{M}^T-M^T{N}^T=NM-MN=-(MN-NM)$ . So,
($MN-NM$) is skew symmetric

(C) $(MN{)}^T=N^T{M}^T=NM\ne MN$ if $M$ and $N$
are symmetric. So, $MN$ is not symmetric

(D) (adj. $M$) (adj. $N$) $=$ adj $\left(NM\right)\ne$ adj ($MN$).