Question

If $A$ and $B$ are symmetric matrices, then $(AB-BA)$ is skew-symmetric.

• A. True
• B. False

Answer 1

The correct option is A True

Let $A$ and $B$ be symmetric matrices.

Then $A^{\prime }=A$ and $B^{\prime }=B$

Now,

$(AB-BA{)}^{\prime }=(AB{)}^{\prime }-(BA{)}^{\prime }$

$=\left(B^{\prime }{A}^{\prime }\right)-\left(A^{\prime }{B}^{\prime }\right)$

$=(BA-AB)$

$=-(AB-BA)$

Thus,

$(AB-BA{)}^{\prime }=-(AB-BA).$

Hence, $(AB-BA{)}^{\prime }$ is skew-symmetric.

Hence, option $A$ is correct.