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

If $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff _________

Solution:

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

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

Now, $A B$ is symmetric if

$(A B)^{T}=A B$

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

$\Rightarrow B A=A B$            [Using (1)]

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

If $A$ and $B$ are symmetric matrices of the same order, then $A B$ is symmetric iff $A B=B A$