Question

For $3\times 3$ matrices $M$ and $N$, which of the following statement(s) is(are) $\text{NOT}$ correct ?

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

Answer 1

Answer: (C), (D)

$\left(adjM\right)\left(adjN\right)=adj\left(MN\right)$ for all invertible matrices $M$ and $N$

$(N^{T}MN{)}^T=N^T{M}^{T}N$
If $M$ is symmetric, then $(N^{T}MN{)}^T=N^{T}MN$
So, $N^{T}MN$ is also symmetric.

If $M$ is skew symmetric, then $(N^{T}MN{)}^T=N^T(-M)N=-N^{T}MN$
So, $N^{T}MN$ is also skew symmetric.


If $M$ and $N$ are symmetric matrices, then
$M^T=M$ and $N^T=N$
$(MN-NM{)}^T$
$=(MN{)}^T-(NM{)}^T$
$=N^T{M}^T-M^T{N}^T$
$=NM-MN$
$=-(MN-NM)$
So, $MN-NM$ is skew symmetric.


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


$\left(adjM\right)\left(adjN\right)=adj\left(NM\right)\ne adj\left(MN\right)$