Question

For $3\times 3$ matrices $M$ and $N$, which of
the following statement(s) is (are) $NOT$ correct?
$I$ $N^{T}MN$ is symmetric or skew-symmetric, according as $M$ is symmetric or skew-symmetric
$II$ $MN-NM$ is skew-symmetric for all symmetric matrices $M$ and $N$
$III$ $MN$ is symmetric for all symmetric matrices $M$ and $N$
$IV$ $\left(adjM\right)\left(adjN\right)=adj\left(MN\right)$ for all invertible matrices $M$ &$N$

• A. $I$ & $II$
• B. $II$ & $III$
• C. $III$ & $IV$
• D. $I$ & $IV$

Answer 1

Answer: (A), (C), (D)

$III$ & $IV$
$I$ & $IV$
$I$ & $II$

A) ${\left(N^{T}MN\right)}^T=N^T{M}^T{\left(N^T\right)}^T=N^T{M}^{T}N$ is symmetric or skew-symmetric, according as $M$ is symmetric or skew-symmetric

B) ${(MN-NM)}^T={\left(MN\right)}^T-{\left(NM\right)}^T=NM-MN=-(MN-NM)$ is skew-symmetric for all symmetric matrices $M$ and $N$

C) ${\left(MN\right)}^T=N^T{M}^T=NM\ne MN$ hence not correct

D) $\left(adjMN\right)=\left(adjN\right)\left(adjM\right)$ hence not correct