So I think I've got a misunderstanding about what curl is or made a calculation mistake. For the vector field $\begin{pmatrix}ax+by\\cx+dy\end{pmatrix}$ then its curl should be $\nabla \times \begin{pmatrix}ax+by\\cx+dy\end{pmatrix} = c-b$ which is a constant. For example $\nabla \times \begin{pmatrix}3x-3y\\-3x\end{pmatrix} = 0$. Which is a constant but the vector field looks like this:
Image of vector field $\begin{pmatrix}3x-3y\\-3x\end{pmatrix}$
This looks like at the bottom left and top right the curl should be positive and at the top and bottom the curl should be negative but the calculated curl is $0$ at all points.
What have I misunderstood about curl?