Answer
$AA^t$ is symmetric.
Work Step by Step
Given : $A$ is symmetric
To Prove: $AA^t$ is symmetric
Proof:
Let us use $(AB)^t=B^tA^t$ and $(A^t)^t=A$
$$(AA^t)^t=(A^t)^tA^t=AA^t$$
Since the transpose of the matrix $AA^t$ results in the matrix $AA^t$ itself.
$AA^t$ is symmetric.