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

Question

Philoid · Accepted Answer

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.