Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent:

- The all-one vector $j$ is contained in the conic hull of $col(A)$.
- The row sums of $A^{-1}$ are non-negative.
- $ADj > 0$, where $D$ is any diagonal matrix with trace $1$.
- The affine hull of $col(-A)$ does not intersect the non-negative orthant.

The equivalency of $(1)$ and $(2)$ follows from the equation $$Ax = j.$$ Assertion $(3)$ is deduced from Farkas' Lemma, as the existence of a positive solution to the above equation implies that there cannot exist a vector $y$ with $y'j = -1$ such that $Ay \geq 0$ (I normalized $y$ without loss of generality). The set of $y$ with sum-of-entries $-1$ is given by $\{y\;|\;y=-Dj: tr(D)=1 \;\text{and}\; D \; \text{diagonal}\}$, the affine combinations of the negative standard basis vectors. This leads to $-ADj <0$.

Finally, the matrix of images of the negative standard basis under $A$ is simply $-A$. Hence, requiring the affine hull of these images not to contain any non-negative vector should be equivalent to $(3)$.

Two sufficient conditions are that $A$ be positive monomial with zero diagonal (as at least one of the entries of $y$ must be negative), or the adjacency matrix of a regular graph. What can be said in general?

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