One of the intriguing and fundamental algorithmic graph problems is the computation of the strongly connected components of a directed graph G. In this paper we first introduce a simple procedure for determining the location of the nonzero elements occurring in $B^{-1}$ without fully inverting B, where EB;{equiv};(b_{ij);and;B^T$ are diagonally dominant matrices with $b_{ii};>;0$ for all i and $b_{ij};{leq};0$, for $i;{ eq};j$, and then, as an application, show that all of the strongly connected components of a directed graph G can be recognized by the location of the nonzero elements occurring in the matrix $C(G);=;(D;-;A(G))^{-1}$. Here A(G) is an adjacency matrix of G and D is an arbitrary scalar matrix such that (D - A(G)) becomes a diagonally dominant matrix.