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Simple recursive methods for inverting an n x n matrix A + E in terms of A-1and E are presented, where E represents a matrix of modifications of the matrix A. Algorithms for rank one and rank r matrix modifications are given. In addition, simple methods for determining if A + E is invertible are developed. Applications of these methods to pattern recognition problems where the inversion of a matrix (eg., covariance matrix, scatter matrix, etc.) must. be computed and frequently updated as changes in data occur are illustrated.