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} $\bar{\mathbf{Z}}=\mathbf{J}-\mathbf{Z}$, where $\mathbf{J}$ is the all-ones matrix. We prove a closed formula for its characteristic polynomial and for its determinant, showing that $\det (\bar{\mathbf{Z}})=(-1{)}^n\tilde{\chi }(P)$, where $n=\mathrm{\mid }P\mathrm{\mid }$ 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.