Question

$A$ and $B$ are symmetric matrices of the same order. $X=AB+BA$ and $Y=AB-BA$.
$(XY{)}^T=$

• A. $XY$
• B. $YX$
• C. $-YX$
• D. $X+Y$

Answer 1

The correct option is C $-YX$

If $A$ and $B$ are symmetric matrices, then $A=A^T$ and $B=B^T$
$X=AB+BA$ $⟹X^T=(AB+BA{)}^T=(AB{)}^T+(BA{)}^T=B^T{A}^T+A^T{B}^T=BA+AB=X$
$Y=AB-BA$ $⟹Y^T=(AB-BA{)}^T=(AB{)}^T-(BA{)}^T=B^T{A}^T-A^T{B}^T=BA-AB=-Y$
$\therefore (XY{)}^T=Y^T{X}^T=-YX$