Question

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

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

Answer 1

The correct option is C

-YX

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