I am reading "Total dual integrality and integer polyhedra" by Giles and Pulleyblank (1979), and I don't understand the proof of theorem 3.2:
Theorem 3.2 For any rational system $\mathbf{Ax}\leq\mathbf{b}$ there exists a rational number $\alpha$ such that $\alpha\mathbf{Ax}\leq\alpha\mathbf{b}$ is TDI.
Proof (with my own notational adaptations): We may assume that $\mathbf{A}$ is $m\times n$ and integer-valued. Let $N$ be the set of all $m\times m$ nonsingular submatrices of the concatenation of $\mathbf{A}$ with the $m\times m$ identity matrix. Let
$$\beta\equiv\left|\prod_{\mathbf{B}\in N}\det(\mathbf{B})\right|\quad\text{and}\quad \alpha\equiv\frac{1}{\beta}.$$
A simple application of Cramer's rule now shows that every component of every basic feasible solution $\mathbf{p}$ of $\mathbf{p}^\text{T}(\alpha\mathbf{A})=\mathbf{c}^\text{T}$, $\mathbf{p}\geq\mathbf{0}$ is integer-valued for any integral $\mathbf{c}$. $\square$
My attempts
(Question 1) So, if I understand correctly, the set $N$ is the set of nonsingular submatrices of this matrix:
$$ [\mathbf{A}\ \ \mathbf{I}\ ] $$
where $\mathbf{I}$ is $m\times m$. Loosely, this seems to correspond to adding slack to the original system $\mathbf{Ax}\leq\mathbf{b}$, but it's not clear to me why we would want to do that.
(Question 2) Next, to write out the proof more formally, it seems like we want to do the following:
Let $\alpha$ be as defined above, let $\mathbf{c}\in\mathbb{Z}^n$, and let $\mathbf{p}$ be a basic feasible solution to $\mathbf{p}^\text{T}(\alpha\mathbf{A})=\mathbf{c}^\text{T}$, $\mathbf{p}\geq\mathbf{0}$. This can be re-written as $\mathbf{p}^\text{T}\mathbf{A}=\beta\mathbf{c}^\text{T}$, $\mathbf{p}\geq\mathbf{0}$. Since $\mathbf{p}$ is a basic solution, it has at most $n$ non-zero entries, stored in a vector $\mathbf{p}_B$, which satisfy
$$\mathbf{B}\mathbf{p}_B = \beta\mathbf{c}$$
where $\mathbf{B}$ is some subset of the rows of the matrix $\mathbf{A}$, such that $\mathbf{B}\in\mathbb{Z}^{n\times n}$. We wish to show that the entries of $\mathbf{p}_B$ are integral. The comment in the paper suggests that we should apply Cramer's rule to this system. Hence, the $i$th component of $\mathbf{p}_B$ is given by
$$\frac{\det(\mathbf{B}_i)}{\det(\mathbf{B})} $$
where $\mathbf{B}_i$ is the matrix obtained by replacing the $i$th column of $\mathbf{B}$ by $\beta\mathbf{c}$.
It is not at all clear to me why this should be an integer. In particular, the quantity $\beta$ was constructed using $m\times m$ determinants, but now it seems that we are interested in $n\times n$ determinants. The wording in the paper is supremely vague, so I'm not sure that I am even applying Cramer's rule to the appropriate system.
Also, I have seen that there are other proofs of this using different techniques. I am specifically interested in the proof outlined in the paper.