If A and B are symmetric matrices, then
(i) $AB - BA$ is a…………
(ii) $BA - 2AB$ is………….

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

Since, ${[AB - BA]^\prime } = {(AB)^\prime } - {(BA)^\prime }$
$= {B^\prime }{A^\prime } - {A^\prime }{B^\prime }$

$= BA - AB$ and $\left. {{B^\prime } = B} \right]$
$= - [AB - BA]$

Therefore, $[AB - BA]$ is a skew-symmetric matrix.

(ii) $[BA - 2AB]$ is a neither symmetric nor skew-symmetric matrix.

$\therefore$ ${(BA - 2AB)^\prime } = {(BA)^\prime } - 2{(AB)^\prime }$

$= {A^\prime }{B^\prime } - 2{B^\prime }{A^\prime }$

$= AB - 2BA$
$= - (2BA - AB)$

Therefore, $[BA - 2AB]$ is neither symmetric nor skew-symmetric matrix.