I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.

Let $A$ be an $l\times n$ matrix with integer entries, $l < n$. Let $K \subseteq \mathbb{R}^n$ denote the kernel of $A$, let $\Delta^{n-1}$ denote the standard simplex in $\mathbb{R}^n$, and let $P = K \cap \Delta^{n-1}$ denote the polytope given by intersecting the kernel of $A$ with the standard simplex. The problem: determine necessary and sufficient conditions for $P$ to be a simplex. For instance, one such condition (assuming $A$ has maximal rank) is that, by Gaussian elimination and permuting columns, $A$ can be expressed in the form $[D C]$ where $D$ is an $l\times l$ diagonal matrix with strictly negative diagonal entries and $C$ is a $l \times (n-l)$ matrix with non-negative entries.

I am sure there are plenty of other equivalent conditions, so any reference that deals with conditions under which the intersection of a simplex and a subspace is a simplex would be appreciated. Thanks.