Question

If $P$ and $Q$ are non singular symmetric matrices and $PQ=QP,$ then

• A. $Q{P}^{-1}\ne P^{-1}Q$
• B. $P^{-1}Q$ is symmetric.
• C. $Q^{-1}P\ne P{Q}^{-1}$
• D. $P{Q}^{-1}$ is non-symmetric.

Answer 1

The correct option is B $P^{-1}Q$ is symmetric.

$PQ=QP$
$\Rightarrow P^{-1}PQ=P^{-1}QP$
$\Rightarrow P^{-1}PQ{P}^{-1}=P^{-1}QP{P}^{-1}$
$\Rightarrow Q{P}^{-1}=P^{-1}Q\dots \left(i\right)$

Now, ${\left(P^{-1}Q\right)}^T=Q^T{\left(P^{-1}\right)}^T$
$=Q{P}^{-1}(\because \text{if}P\text{is symmetric, then}{P}^{-1}\text{is also symmetric})$
$=P^{-1}Q(\text{from}\left(i\right)\text{)}$
$\Rightarrow P^{-1}Q$ is symmetric.

Similarly, $Q^{-1}P=P{Q}^{-1}$
and $P{Q}^{-1}$ is also symmetric.