Question

If $A$ and $B$ are symmetric matrices, prove that $AB-BA$ is a skew symmetric matrix.

Answer 1

It is given that $A$ and $B$ are symmetric matrices.

Therefore, we have:

$A^{\prime }=A$ and $B^{\prime }=B$..............(1)

Now, ${(AB-BA)}^{\prime }={\left(AB\right)}^{\prime }-{\left(BA\right)}^{\prime },[\because {(A-B)}^{\prime }=A^{\prime }-B^{\prime }]$

$=B^{\prime }{A}^{\prime }-A^{\prime }{B}^{\prime },[\because {\left(AB\right)}^{\prime }=B^{\prime }{A}^{\prime }]$

$=BA-AB$ ................ [using (1) ]

$=-(AB-BA)$

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

Thus, $(AB-BA)$ is a skew-symmetric matrix.