Question 26: Let n and N be positive integers, and let \mathcal{F} be…

Let $n$ and $N$ be positive integers, and let $\mathcal{F}$ be an arbitrary nonempty family of subsets of the universe $\{1, \ldots, n\}$. Suppose each element $x \in \{1, \ldots, n\}$ in the universe receives an integer weight $w(x)$, each of which is chosen independently and uniformly at random from $\{1, \ldots, N\}$. The weight of a set $S$ in $\mathcal{F}$ is defined as

$w(S) = \sum_{x \in S} w(x)$

We want to explore the probability of the following "good" event: there is a unique set in $\mathcal{F}$ that has the minimum weight among all sets of $\mathcal{F}$.

Based on the above data, answer the given subquestions.

We continue the notation from the previous question. Suppose the randomly assigned weights lead to the good event, that is, there is an unique perfect matching in G with minimum weight, and say this minimum weight is r. What can you say about the determinant of Z in this case?