The eccentricity matrix E(G) of a simple connected graph G is obtained from the distance matrix D(G) of G by retaining the largest distance in each row and column, and by defining the remaining entries to be zero. This paper focuses on the eccentricity matrix \(E(W_n)\) of the wheel graph \(W_n\) with n vertices. By establishing a formula for the determinant of \(E(W_n)\) , we show that \(E(W_n)\) is invertible if and only if \(n \not \equiv 1\ (\textrm{mod}\ 3)\) . We determine the inertia of \(E(W_n)\) by obtaining the determinant and inertia of the eccentricity matrix of the fan graph. Further, we derive a formula for the inverse of \(E(W_n)\) by finding a vector \(\textbf{w}\in \mathbb {R}^n\) and an \(n \times n\) symmetric Laplacian-like matrix \(\widetilde{L}\) of rank \(n-1\) such that \(\begin{aligned} E(W_n)^{-1} = -\frac{1}{2}\widetilde{L} + \frac{6}{n-1}\textbf{w}\mathbf {w^{\prime }}. \end{aligned}\)
