Distributing matrix-vector product across a element-wise (hadamard) product

Question

Given this equation, how do you solve for $c_3$ (in terms of $M$, $c_1$ and $c_2$)?

$(Mc_1 + e_1) \odot (Mc_2 + e_2) = Mc_3 + e_3$

I'm not sure how to algebraically distribute across the hadamard product, and trying to expand it out left me with some strange terms I couldn't figure out how I got there.

Answer 1

Let $x_k = Mc_k+e_k$
Then $$\eqalign{ x_3 &= Mc_3+e_3 = x_1\odot x_2 \cr Mc_3 &= x_1\odot x_2 - e_3 \cr c_3 &= M^{-1}(x_1\odot x_2 - e_3) \cr\cr }$$