Let $G$ be a finite abelian group, and let $\omega: G \times G \to \mathbb{C}^\times$ be a symmetric group pairing, i.e. a map inducing a group homomorphism $\tilde{\omega}:G \to G^*$. We say that $\omega$ is non degenerate iff $\tilde{\omega}$ is an isomorphism. In case $G=\mathbb{Z}/n$, $\omega$ is (i) uniquely defined by a $n$th root of unity $\omega(\chi,\chi)=e^{\frac{2 \pi i \cdot k}{n}}$, $0<k \leq n$ and (ii) non-degenerate iff $gcd(k,n)=1$. What are the corresponding statements for $G=\mathbb{Z}/n \times \mathbb{Z}/n$?
Edit: I know that $\omega$ is uniquely defined by a $2\times 2$-matrix $A$ with $\mathbb{Z}/n$-coefficients, where $\omega(\chi_i,\chi_j)=\text{exp}\left(\frac{2 \pi i \cdot A_{ij}}{n}\right)$. The question is, when this is non-degenerate. It seems that the condition is $\text{det}(A) \neq 0$.