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

Question

Philoid · Accepted Answer

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

Symmetric matrix: A symmetric matrix is a square matrix that is equal to its transpose. In simple words, matrix A is symmetric if

A = A’

where A’ is the transpose of matrix A.

Skew Symmetric matrix: A skew symmetric matrix is a square matrix that is equal to minus of its transpose. In simple words, matrix A is skew symmetric if

A = -A’

Given: A and B are symmetric matrices i.e.

A = A’ …(1)

B = B’ …(2)

Now calculating the transpose of AB – BA,

(AB – BA)’ = (AB)’ – (BA)’

(By property of transpose i.e. (A – B)’ = A’ – B’)

= B’A’ – A’B’

(By property of transpose i.e. (AB)’ = B’A’)

= BA – AB

= -(AB – BA)

Or we can say that: (AB – BA) = - (AB – BA)’

Clearly it satisfies the condition of skew symmetric matrix.

Hence AB – BA is a skew symmetric matrix.