# Bounding the Solution of an Integral Matrix Equation

What are the necessary and sufficient condition(s) on a non-negative integral, invertible square $n \times n$ matrix $A$ to ensure that the unique solution of the matrix equation $A \mathbf{x} = \mathbf{1}$ lies in the interval $(0, \tfrac{1}{2}]^{n}$? Of course, Cramer's rule implies the condition $0 < 2 \det A_{j,1} \leq \det A$ for $1 \leq j \leq n$, where $A_{j,1}$ is the matrix $A$ with the $j^{\text{th}}$-column replaced with $\mathbf{1}$.

Farkas' Lemma gives conditions for positivity, but it is usually reserved for real matrices and seems a bit high powered for my question.

Thanks!

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