Q. If $P$ and $Q$ are symmetric matrices of the same order then $PQ - QP$ is

Solution:

Given $P = P'$ and $Q=Q'$
$\left(PQ-QP\right)'=\left(PQ\right)'-\left(QP\right)^{'}$
$=\left(Q'P'- P'Q'\right)=QP-PQ$
$-\left[PQ-QP\right]$
$\therefore\, \left(PQ-QP\right)$ is skew symmetric.

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