TITLE: Invertibility of combinatorial matrix
QUESTION [6 upvotes]: Fix integers $n$ and $k$ with $0 \leq k < \dfrac{n}{2}$. Let $I$ be the set of all $k$-element subsets of $\{1,\ldots,n\}$. Consider the matrix $A=(a_{ij})_{i,j\in I}$ where $a_{ij}=1$ if $i\cap j=\emptyset$ and $a_{ij}=0$ otherwise. What is the easiest way to see that $\det(A)\neq 0$?

For example, for $n = 3$ and $k = 1$, we have $\det(A) = 2$. More generally, for $k = 1$, we have $\det(A) = \pm (n-1)$ (since $A$ is the all-ones matrix minus the identity matrix).

REPLY [6 votes]: Your matrix is the adjacency matrix of the Kneser graph $KG_{n, k}$ (using the notation from the Wikipedia article). Its eigenvalues are known; here is a quote from the Wikipedia:

For $j=0,\ldots,k$, the eigenvalue $\lambda_j=(-1)^j\dbinom{n-k-j}{k-j}$ occurs with multiplicity $\dbinom{n}{j}-\dbinom{n}{j-1}$ for $j > 0$ and $1$ for $j = 0$. See this paper for a proof.

This is also proved in

Corollary 6.6.1 of Chris Godsil, Karen Meagher, Erdös–Ko–Rado Theorems: Algebraic Approaches, Cambridge University Press 2016.
Theorem 1 of Philipp Reinfeld, Chromatic Polynomials and the Spectrum of the Kneser Graph (freely available).

The oft-given reference to Godsil and Royle, Algebraic Graph Theory seems wrong: they only appear to prove the set of eigenvalues, not their multiplicities. (Although the multiplicities might follow from their argument, if there is a way to lift them through an equitable partition.)
Note that the words "with multiplicity $\dbinom{n}{j}-\dbinom{n}{j-1}$ for $j > 0$ and $1$ for $j = 0$" can be replaced by "with multiplicity $\dbinom{n}{j}-\dbinom{n}{j-1}$", since $\dbinom{n}{j}-\dbinom{n}{j-1}$ equals $0$ when $j = 0$.
As you know, the determinant of a matrix is the product of its eigenvalues. Thus, the above formula for the eigenvalues yields that the determinant of your matrix is
\begin{align}
\prod_{j=0}^k \left(\left(-1\right)^j \dbinom{n-k-j}{k-j}\right)^{\dbinom{n}{j}-\dbinom{n}{j-1}} \neq 0
\end{align}
(since all $\dbinom{n-k-j}{k-j}$ are positive integers when $k \leq n/2$).