The paper deals with matrices over an additively idempotent semiring \(L\) (path algebra). A path algebra 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 author generalizes many previous results concerning matrices over these particular algebraic structures. The main results bring a characterization for matrices of finite order, upper bounds of matrix index and period, formulas for transitive closure, reduced matrices and a description of the family of all eigenvectors (case \(\lambda =1\)). Some results assume that the semiring \(L\) is additively residuated. The paper contains a rich list of references to generalized results. Reviewer's remark: 1) The name `almost periodic element' is used by the author instead of `element of finite order' (name commonly used in semigroup theory, cf. e.g. [\textit{J. M. Howie}, Fundamentals of semigroup theory. Oxford: Clarendon Press (1995; Zbl 0835.20077)]). 2) In section 3 the author reproves some known properties of ordered semigroups (cf. e.g. [\textit{J. Drewniak} and \textit{J. Sobera}, Czech. Math. J. 53, No.~4, 777--791 (2003; Zbl 1080.06019)]).