Question

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

Answer 1

For the symmetric matrices A and B ,

A'=A B'=B }

Now,

( AB−BA )'=( AB )'−( BA )'      [ ∵( A−B )'=A'−B' ] =B'A'−A'B'        [ ∵( AB )'=B'A' ] =BA−AB =−( AB−BA )

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