Putnam 2018 A2

Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some order, and let $M$ be the $(2^n-1) \times (2^n-1)$ matrix whose $(i,j)$ entry is $$m_{ij} = \begin{cases} 0 & \text{if }S_i \cap S_j = \emptyset; \\ 1 & \text{otherwise.} \end{cases}$$ Calculate the determinant of $M$.