Question

If A and B are symmetric matrices of the same order, then AB is symmetric iff ______________.

Answer 1

It is given that, A and B are symmetric matrices of the same order.

$\therefore A^T=A$ and $B^T=B$ .....(1)

Now, AB is symmetric if

${\left(AB\right)}^T=AB$

$\Rightarrow B^T{A}^T=AB$

$\Rightarrow BA=AB$ [Using (1)]

Thus, if A and B are symmetric matrices of the same order, then AB is symmetric iff AB = BA.

If A and B are symmetric matrices of the same order, then AB is symmetric iff __AB = BA__.