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. Ior — I, where I is an identity matrix.

Answer 1

Answer: (B)

$A=A^T,B=-B^T$
Let $P=AB-BA$
$P^T=(AB-BA{)}^T$
$=(AB{)}^T-(BA{)}^T$
$=B^T{A}^T-A^T{B}^T$
$=-BA+AB$
$=AB-BA$
$=P$
So $AB-BA$ is symmetric