This paper derives a set of new algorithms based on the exchange method for the computation of matrix inverses including nonsingular, symmetric nonsingular, and rectangular matrices. The symmetric matrix inversion algorithm can save up to 50 percent of the computation time required for the Gauss-Jordan elimination method. The pseudoinverse algorithms derived here are very attractive in terms of small storage requirement, short computation time, and high numerical accuracy. Comparisons are made between the new algorithms and existing ones, and numerical examples are included.