Question

P- symmetric matrix, Q- skew symmetric matrix, P and Q are square matrice of same order, then what type of a matrix is PQ-QP

Answer 1

The product of a symmetric matrix and a skew symmetric matrix is skew symmetric matrix

The product of a skew symmetric matrix and a symmetric matrix is skew symmetric matrix

So$\left(PQ\right)=(-QP{)}^T$ and $QP=(-PQ{)}^T$ Hence

$PQ-QP=--(QP{)}^T+(PQ{)}^T=(PQ-QP{)}^T$

Proving that $PQ-QP$ is symmetric.