Given two matrices $A, B \in \mathbb{R}^{n \times m}$, where $A = \{a_{i,j}\}$ and $B = \{b_{i,j}\}$, the Hadamard product (or point-wise product) is a matrix $$C = A \circ B$$ such that $C = \{c_{i,j}\} = \{a_{i,j} \cdot b_{i,j}\}$.
I'm not so skilled with this operator. I googled around and I found few things, just definitions and some relevant properties.
At the moment, I'm working on this kind of problem:
Consider $A, B \in \mathbb{R}^{N \times N}$ and $C \in \mathbb{R}^{N \times N}_{\geq 0}$. Find $x \in [0,1]^N$ such that $$M x = Qy,$$
where $M = (A+B)\circ C$, $Q=B \circ C$ and $y = [1 ~ 1 \ldots 1]^\top \in \mathbb{R}^{N}$.
Of course, I'm looking for the condition on matrices $A$, $B$ and $C$ such that one can solve the previous problem.
It can be also useful to solve a relaxed version of the problem:
Consider $A, B \in \mathbb{R}^{N \times N}$ and $C \in \mathbb{R}^{N \times N}_{\geq 0}$. Find $x \in \mathbb{R}^N$ such that $$M x = Qy,$$
where $M = (A+B)\circ C$, $Q=B \circ C$ and $y = [1 ~ 1 \ldots 1]^\top \in \mathbb{R}^{N}$.
I was able to find some condition for the existence of a solution $x$ when $A$ and $B$ have constant rows. But I would like to find general conditions.
The problem is too broad, and therefore I would like just some hints for the case $C \in \mathbb{R}^{N \times N}_{\geq 0}$, i.e. $c_{i,j} \geq 0 ~\forall i, j$.
