Let $A$ be a regular language. Consider the following operations on $A$:
$2A:=\{xy \mid x, \: y \in A \text{ and } x=y\}$
$A^2 :=\{xy \mid x, \: y \in A\}$
One of these operations necessarily leads to a regular language and the other may not. Identify which is which. For the regular operation, give a proof that it is regular. For the non-regular operation, give an example of an $A$ such that applying the operation on it results in a non-regular language.