Question

If A and B are symmetric matrices, then write the condition for which AB is also symmetric.

Answer 1

Given:  AB is symmetric.

`⇒( AB) ^T = AB`

`⇒ B^T A^T = AB            [∵( AB)^T = B^T A^T ]`

`⇒ BA = AB [ A and \text{B are symmetric matrices},  \text{ so } A^T= A and B^T = B]` 

Thus, AB is also symmetric, if AB = BA.