Question

Let $A$ and $B$ be two symmetric matrices of same order. If $X=AB+BA$ and $Y=AB-BA,$ then $(XY{)}^T$ is equal to

• A. $YX$
• B. $-XY$
• C. $XY$
• D. $-YX$

Answer 1

The correct option is D $-YX$

$A$ and $B$ are two symmetric matrices.
$\Rightarrow A^T=A$ and $B^T=B$
$(XY{)}^T=Y^T{X}^T$

$X^T=(AB+BA{)}^T$
$=(AB{)}^T+(BA{)}^T$
$=B^T{A}^T+A^T{B}^T$
$=BA+AB=X$
Similarly, $Y^T=(AB-BA{)}^T=BA-AB=-Y$

Hence, $(XY{)}^T=-YX$