I have a matrix equation of the form:

$$ A^{-1} = B + A \circ C $$

where $\circ$ denotes the Hadamard product (i.e., $(A\circ C)_{ij} = A_{ij}B_{ij}$). How can I determine if a solution for $A$ exists, if it's unique, and ultimately solve for $A$?

Edit: I'm only interested in the case of real valued matrices.