I have to deal with the following linear equation system:
\begin{align} &\boldsymbol{A} \mathbf{x} = \mathbf{b}, \text{ s.t.} \\ &\boldsymbol{A} \in \mathbb{R}_{\geq0}^{n \times n}, \forall i,j \in \{1, \dots, n\}: A_{ii} = 0, A_{ij} > 0 \text{ if } i \neq j,\\ &\boldsymbol{x} \in \mathbb{R}^n_{\geq0},\\ &\mathbf{b} = \begin{bmatrix} c \\ \vdots \\ c \end{bmatrix} \in \mathbb{R}_{>0}^n. \end{align}
My question is: Which properties must matrix $\mathbf{A}$ have such that we obtain a valid solution $\mathbf{x} \geq \mathbf{0}$ for the above described linear equation system?
For $n=1$, no solution exists. For $n=2$, there always exists a solution, namely $x_1 = \frac{c}{A_{21}} > 0$ and $x_2 = \frac{c}{A_{12}} > 0$.
I am looking forward to reading your replies.