Question

If $A$ and $B$ are symmetric matrices of the same order the $(AB-BA)$ is always.

• A. A symmetric matrix
• B. A skew-symmetric matric
• C. A zero matrix
• D. An identity matrix

Answer 1

Answer: (B)

A skew-symmetric matric

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

Formulas used

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

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

$\therefore (AB-BA{)}^{\prime }=(AB{)}^{\prime }-(BA{)}^{\prime }=(B^{\prime }{A}^{\prime }-A^{\prime }{B}^{\prime })=(BA-AB)=-(AB-BA)$
$\therefore (AB-BA)$ is skew symmetric.