Given a vector field $F = Ax$, where $A \in \mathbb R^{2 \times 2}$ and $x \in \mathbb R^2$, find the curl of $F$.
In this case, if we know that the matrix is a constant is it right for me to conclude that it is a constant vector field and hence it has a $div F = 0$ and $curl F = 0$? How to provide a proof for this?
No, the vector field is not constant. The vector field, if the matrix is
$$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$
is equal to $$f(x,y)=(ax+by, cx+dy)$$
which is pretty far from being constant...