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This paper presents a parallel algorithm for obtaining the inverse of a large, nonsingular symmetric matrix A of dimension nxn. The inversion method proposed is based on the triangular factors of A. The task of obtaining the "sparse inverse' of A is represented by a directed acyclic graph. The relation between the triangulation graph and the sparse inversion graph is given. The algorithm and the graph for the full inversion of A is also given. It is shown that for any sparse symmetric matrix, and assuming enough processors are available, the full inverse of the matrix can be calculated in the same amount of time as the sparse inverse. For ideally sparse matrices (such as tridiagonal matrices) the order of computation required in both cases is of order log2n. For full matrices the order of computation is n log2n. Claims are substantiated using test data from several power systems.