Question

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

• A. zero matrix
• B. identity matrix
• C. skew symmetric matrix
• D. symmetric matrix

Answer 1

Answer: (C)

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