Question

If A and B are symmetric matrices, such that AB and BA are both defined, then prove that AB- BA  is a skew-symmetric matrix.

Answer 1

Since A & B are symmetric matrices A = A’ & B = B’ 

To prove (AB – BA)’ = – (AB – BA) 

L.H.S = (AB – BA)’ 

(AB)’ – (BA)’ 

= B’A’-A’B’ 

= BA – AB 

= – (AB – BA) 

= R.H.S 

Thus proved.