Question

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

• A. Skewsymmetric
• B. Symmetric
• C. Neither symmetric nor skewsymmetric
• D. I or -I, where I is an identity matrix.

Answer 1

The correct option is B Symmetric

${(AB-BA)}^T=(AB{)}^T-(BA{)}^T$

$=B^T{A}^T-A^T{B}^T$

$=-BA+AB$

${(AB-BA)}^T=AB-BA$ .

$AB-BA$ is symmetric.