A.V. Arhangelskii introduced the dimension Dind and various studies have been devoted to this topological dimension, investigating many of its properties, like the closed subspace and union theorems, and its behaviour in the class of all finite \(T_0\) -spaces. In this paper, we extend these studies, investigating the dimension Dind for finite \(T_0\) -spaces via Matrix Algebra. We give firstly new topological characterizations of Dind through minimal open sets, inserting new notions of families consisting of such open sets. Also, since the dimension Dind is an inductive dimension, based on the dimension Dind of special closed subsets, we study a matrix approach of these subsets, presenting an algorithm which finds their incidence matrices. The above investigations lead to new characterizations of Dind through incidence matrices, which finally provide an algorithmic procedure for the computation of Dind for finite \(T_0\) -spaces.
