Let $R=\mathbb{C}[x,y]$ and let $\phi: R^2 \to R^2$ be the $R$-module homomorphism given by the matrix $A= \left( \begin{array}{ccc} a & b \\ c & d \\ \end{array} \right) $ where $a,b,c,d \in \mathbb{C}[x,y]$.
What are the necessary and sufficient conditions for $\phi$ to be an isomorphism?
I know $\phi$ will be bijective iff $A$ has an inverse iff $ad-bc \neq 0$.
Your matrix $A$ will have inverse iff $\det A$ has inverse in $R=\mathbb C[x,y]$. Now, the invertible elements of $R$ are the invertible in $\mathbb C$, that is, $\mathbb C\setminus\{0\}$. Let us see why.
If $p(x,y)$ is invertible in $R$, then exists $q(x,y)$ so that $p(x,y)\cdot q(x,y)=1$. But that would mean $$\deg p(x,y) + \deg q(x,y) = \deg [p(x,y)\cdot q(x,y)]= \deg 1=0$$ so $\deg p(x,y) = \deg q(x,y) = 0$. That means $p(x,y)\in \mathbb C$.
So, to sum up, $\phi$ is an isomorphism iff $\det A \in \mathbb C\setminus \{0\}$.