If A and B are symmetric matrices of the same order, write whether AB – BA is symmetric or skew-symmetric or neither of the two.

Question

Philoid · Accepted Answer

A and B are symmetric matrices,

∴ A’ = A and B’ = B …(i)

Consider (AB – BA)’ = (AB)’ – (BA)’ [(a – b)’ = a’ – b’]

= B’A’ – A’B’ [(AB)’ = B’A’]

= BA – AB [from (i)]

= - (AB – BA)

∴ (AB – BA)’ = - (AB – BA)

Hence, (AB – BA) is a skew symmetric matrix.