If A, B are symmetric matrices of same order, then AB – BA is a


Given A and B are symmetric matrix of same order

A = A’ eqn1


B = B’ eqn2


So, AB – BA = A’B’ – B’A’ (from 1 & 2)


AB –BA = (BA)’ – (AB)’ ()


AB – BA = (-1) ((AB)’ – (BA)’) (taking -1 common)


AB – BA = -(AB – BA)’ ()


Here we see that the relation between (AB – BA) and its transpose i.e. (AB – BA)’ is (AB –BA) = -(AB – BA)’, this implies that (AB – BA is a skew symmetric matrix.


Hence proved.

11
1