Question

Suppose $A$ and $B$ are two square matrices of same order. If $A,B$ are symmetric matrices, then $AB-BA$ is

• A. A symmetric matrix
• B. A skew symmetric
• C. A scalar matrix
• D. A triangular matrix

Answer 1

The correct option is B A skew symmetric

${(AB-BA)}^T={\left(AB\right)}^T-{\left(BA\right)}^T=B^T{A}^T-A^T{B}^T=BA-AB=-(AB-BA)$

This shows that $(AB-BA)$ is a skew-symmetric matrix