The paper deals with matrices over an additively idempotent semiring (path algebra). A path algebra is an important tool e.g. in automata theory, combinatorial optimization and switching circuits. It is a generalization of many algebraic structures as e.g. Boolean algebra, fuzzy algebra, De Morgan algebra, max-plus algebra, min-plus algebra or incline algebra. The goal of the paper is an examination of nilpotent matrices over the path algebra enriched by the existence of additive residuals (semiring difference). We get an extended version of a recent author's paper (common with \textit{M. Y. Guan}) [J. Fuzhou Univ., Nat. Sci. 37, No.~2, 157--161 (2009; Zbl 1212.15039)]. The paper brings a precise characterization of nilpotency and describes many of its consequences (Theorems 3.1, 3.2). Additionally, properties of a transitive closure are compared for a matrix \(A\) and its reduction \(A-A^2\) using the semiring difference (Theorems 4.1--4.3). This provides generalizations of many previous results concerning fuzzy matrices, lattice matrices and incline matrices [cf. \textit{H. Hashimoto}, Inf. Sci. 27, 233--243 (1982; Zbl 0524.15014); \textit{K.-L. Zhang}, Fuzzy Sets Syst. 117, No.~3, 403-406 (2001; Zbl 0971.15008); \textit{Yi-jia Tan}, Fuzzy Sets Syst. 151, No.~2, 421--433 (2005; Zbl 1062.06021); \textit{S.-C. Han, H.-X. Li} and \textit{J.-Y. Wang}, Linear Algebra Appl. 406, 201--217 (2005; Zbl 1082.15040)].