Question

Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix

Answer 1

Let A and B be two symmetric matrices

⇒ A^T = A and B^T = B    ........(1)

Given that AB = BA (2)

To prove AB is symmetric:

Now (AB)^T = B^TA^T = BA

(From(1)) But (AB)^T = AB by     ........(2)

⇒ AB is symmetric.

Conversely let AB be a symmetric matrix.

⇒ (AB)^T = AB

i.e. B^TA^T = AB

i.e. BA = AB  .......(From (1))

⇒ AB is symmetric