if 3 SAT is reduced to A, as well as A is reduced to 3SAT, then is A in NPC?
A polynomial from $3SAT$ to $A$ shows that $A$ is $NP$-hard.

A polynomial reduction from $A$ to $3SAT$ shows that $A$ is in $NP$.

So, the answer is yes.

You can construct a polynomial verification algorithm for $A$ by first reducing $A$ to $3SAT$, and then using the verification algorithm for $3SAT$.