Counting sums of exceptional units in ${\mathbb{Z}}_n$

Let $R$ be a commutative ring with the identity $1_R$, and let $R^{\ast }$ be the multiplicative group of units in $R$. An element $a\in R^{\ast }$ is called an exceptional unit if there exists a $b\in R^{\ast }$ such that $a+b=1_R$. We set $R^{\ast \ast }$ to be the set of all exceptional units in $R$. In this paper, we consider the residue-class ring ${\mathbb{Z}}_n$. For any positive integers $n,s$, and $c\in {\mathbb{Z}}_n$, let ${\mathcal{N}}_s(n,c):=\mathrm{♯}\{(x_1,...,x_s)\in ({\mathbb{Z}}_n^{\ast \ast }{)}^s:x_1+...+x_s\equiv c(\mathrm{mod}n)\}$. In 2016, Sander (J.Number Theory 159 (2016)) got a formula for ${\mathcal{N}}_2(n,c)$. Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander's theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for ${\mathcal{N}}_s(n,c)$. This extends and improves earlier results.