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

The correct option is C Skew symmetric matrix

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

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

So $PQ=-(QP{)}^{T}andQP=-(PQ{)}^T$. Hence

$PQ-QP=-(QP{)}^T+(PQ{)}^T=(PQ-QP{)}^T$, proving that $PQ-QP$ is a skew symmetric matrix.