Question

If A and B are symmetric matrices of same order, prove that AB – BA is a skew-symmetric matrix

Answer 1

Given A and B are symmetric matrices

⇒ – A^T = A and B^T = B

To prove AB – BA is a skew-symmetric matrix.

Proof: (AB – BA)^T = (AB)^T – (BA)^T 

= B^TA^T – A^TB^T 

= BA – AB

i.e. (AB – BA)^T = – (AB – BA)

⇒ AB – BA is a skew symmetric matrix.