For $n\ge 1$, ``n-particle zeon algebra'' ${\mathcal{C}\ell_n}^{\rm nil}$ is the $2^n$-dimensional real commutative algebra generated by the unit scalar $1$ along with the ``zeons'' $\{\zeta_i: 1\le i \le n\}$ satisfying the zeon commutation relations \begin{equation*} \zeta_i\zeta_j=\begin{cases} 0&\text{if and only if }i=j\\ \zeta_j\zeta_i&\text{otherwise}. \end{cases} \end{equation*} Any element ${u\in\mathcal{C}\ell_n}^{\rm nil}$ can be written as the sum of a scalar (real) part $\mathfrak{R}u$ and a nilpotent (dual) part, $\mathfrak{D}u$. Utilizing multi-index notation, $u$ has canonical expansion of the form \begin{equation*} u=\sum_{I\subseteq[n]}u_I\zeta_I, \end{equation*} where $[n]=\{1, 2,\ldots, n\}$. The \emph{zeon-Hodge dual} of $u$, denoted $u^\star$, is defined by \begin{equation*} u^\star=\sum_{I\subseteq[n]}u_I\zeta_{[n]\setminus I}. \end{equation*} In the current work, elementary properties of matrices having entries in ${\mathcal{C}\ell_n}^{\rm nil}$ are developed and some applications for special cases are considered. For example, let $A=(a_{ij})$ be a square matrix having entries from ${\mathcal{C}\ell_n}^{\rm nil}$, and write $A=\mathfrak{R}A + \mathfrak{D}A$, where $\mathfrak{R}\mspace{1mu} A=(\mathfrak{R}a_{ij})$. It follows that $A$ is invertible if and only if $\mathfrak{R}\mspace{1mu} A$ is invertible and moreover, the inverse is given by \begin{equation*} A^{-1}=(\mathfrak{R}\mspace{1mu} A)^{-1}\sum_{\ell=0}^{\kappa(\mathfrak{D}A(\mathfrak{R}\mspace{1mu} A)^{-1})-1}(-1)^\ell(\mathfrak{D}A(\mathfrak{R}\mspace{1mu} A)^{-1})^{\ell}, \end{equation*} where $\kappa(M)$ denotes the \emph{index of nilpotency} of matrix $M$. As an application of invertible zeon matrices, begin by recalling that the nilpotent adjacency matrix $\Phi=(\varphi_{ij})$ associated with a simple graph $G=(V,E)$ on $n$ vertices is an $n\times n$ matrix having entries in ${\mathcal{C}\ell_n}^{\rm nil}$ defined by \begin{equation*} \varphi_{ij}=\begin{cases} \zeta_j &\text{if }\{v_i, v_j\}\in E\\ 0&\text{otherwise.} \end{cases} \end{equation*} Extending the zeon Hodge dual to matrices with zeon entries, the \emph{zeon conjugate} of a matrix $A=(a_{ij})$ is defined to be the zeon Hodge dual-transpose of $A$; i.e., $A^\dagger=(\alpha_{ij})$ where $\alpha_{ij}=a_{ji}^\star$. Given a simple graph $G$ on $n$ vertices having nilpotent adjacency matrix $\Phi$, the \emph{zeon Laplacian} of $G$ is defined herein as the $n\times n$ matrix $\Lambda=(\Phi^\dagger \Phi-\Phi^\dagger)^\dagger$. With modified Dirac notation, the following theorem is established. Theorem. When $\Lambda$ denotes the zeon Laplacian of a simple graph $G$ on $n$ non-isolated vertices, the following holds: \begin{eqnarray*} \langle \zeta_{\{i\}}\vert \Lambda^{-1}\vert v_j\rangle&=&\sum_{k=1}^n \sum_{\{I: |I|=k+1\}}\frac{1}{\deg(v_I)}\omega_I\zeta_I, \end{eqnarray*} where $\omega_I$ denotes the number of paths of length $k$ from vertex $v_i$ to vertex $v_j$ on vertices indexed by $I$, and $\deg(v_I)$ denotes the product of vertex degrees over vertices indexed by $I$.