Complexity of an instance of a NP-complete problem
To show $A$ is NP-complete, you need to prove the following: 


$A \in \mathbf{NP}$
$A$ is NP-hard: any problem $B \in \mathbf{NP}$ is poly-time (many-one) reducible to $A$

or

equivalently (because reduction relations are transitive): there is an NP-hard problem $B$ which is poly-time (many-one) reducible to $B$. 


You have shown neither, so you cannot say $A$ is NP-complete.

What you did do is a many-one reduction from $A$ to $B$, but you want the other way around. In addition, you also need to show the reduction is efficient, that is, takes polynomial time; you did not explicitly state whether this is the case. And, of course, you must not forget to show $A \in \mathbf{NP}$ (which is usually the easier part).