Question

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

Answer 1

Here, A and B are symmetric matrices, then A′ = A and B′ = B 

Now, (AB − BA)′ = (AB)′ − (BA)′ (∵(A − B)′ = A′ − B′ and (AB)′ = (B′A′) 

= B ′A′ − A′B′ = BA − AB (∵ B′ = B and A′ = A) 

= −(AB − BA) 

⇒(AB − BA)′ = −(AB − BA) 

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