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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1 Answer

The inequality you derived from Cramer's rule is both necessary and sufficient. You can see this by verification (or disconfirmation) of the hypothesis on each component individually.

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Perhaps I should rephrase the question. Are there any other necessary and sufficient conditions which ensure the bound, perhaps extensions of special cases of Farkas' Lemma or the like? – user02138 Jun 17 '11 at 3:50
They would be logically and mathematically equivalent to your Cramer-based system of inequalities. It seems you might either be seeking a computationally more appealing form, or an alternate expression that is more usable for some other mathematical purposes. I'm uncertain as to how to look for or if there's a way to transform the conditions you have into a nontrivially distinct-looking alternative. – anon Jun 17 '11 at 3:58