Question

Let $A$ and $B$ be any two $3\times 3$ matrices. If $A$ is symmetric and $B$ is skew-symmetric, then the matrix $AB-BA$ is always :

• A. Skew symmetric
• B. Symmetric
• C. Neither symmetric nor skew symmetric
• D. Lower triangular matrix

Answer 1

The correct option is B Symmetric

$(AB-BA{)}^T=(AB{)}^T-(BA{)}^T$
$=B^T{A}^T-A^T{B}^T$
$=-BA+AB$
$=AB-BA$
$\Rightarrow AB-BA$ is symmetric.