Question

If $A$ and $B$ are symmetric matrices of the same order and $X=AB+BA$ and $Y=AB-BA,$ then $X{Y}^T$ is equal to

• A. $XY$
• B. $YX$
• C. $-XY$
• D. None of these

Answer 1

The correct option is C $-XY$

Given, $A$ and $B$ are symmetric.
Then, $A^T=A$ and $B^T=B$
$X{Y}^T=(AB+BA)(AB-BA{)}^T$
$=(AB+BA)\left(\right(AB{)}^T-\left(BA{)}^T\right)$
$=(AB+BA)(B^T{A}^T-A^T{B}^T)$
$=(AB+BA)(BA-AB)$
$=-(AB+BA)(AB-BA)$
$=-XY$
$\therefore X{Y}^T=-XY$