Let $Y$ be a member of $\mathbb{R}^m$. I need a necessary and sufficient condition on a $n\times m$ binary matrix $A$ for having a solution to the linear equation: $$AX=Y$$ Such that $X_i\geq 0$, $\forall i\in\{1,\dots,m\}$.
Do you know a necessary and sufficient condition for unicity of this "positive" solution?
check M-matrices:
http://en.wikipedia.org/wiki/M-matrix
If $A$ is an invertible M-matrix, then $A^{-1}$ has non-negative entries. That is, if $y\ge 0$, then the solution $x$ to $Ax=y$ is $x\ge0$.