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