A formula for the Euler characteristic of a poset through the determinant of the order-complement matrix

Given a finite poset $P$, its zeta matrix $\mathbf Z$ encode fundamental incidence-theoretic information about the order structure. In this paper we introduce and study the \emph{order-complement matrix} $\overline{\mathbf Z} = \mathbf J - \mathbf Z$, where $\mathbf J$ is the all-ones matrix. We prove a closed formula for its characteristic polynomial and for its determinant, showing that $\det(\overline{\mathbf Z}) = (-1)^n \tilde{\chi}(P)$, where $n = |P|$ and $\tilde{\chi}(P)$ is the reduced Euler characteristic of $P$. This provides a new, unexpectedly simple linear-algebraic expression for the Euler characteristic of a poset, complementing existing determinant formulas for matrices derived from incidence relations.