An important problem in econometric theory requires inversion of a Leontief matrix of inter-industry transactions, the order of such a matrix being equal to the number of industries comprising the given economy. For Leontief matrices of high order, to avoid accumulation of round-off errors inherent in direct processes of matrix inversion such as the Gauss-Doolittle method, F. V. Waugh [11] proposed a certain power series (Neuman's Series) which approximates the inverse of a Leontief matrix to any desired degree of accuracy. The present paper indicates several other power series inversions of the Leontief matrix each of which converges more rapidly than the series roposed by Waugh. Application of the new method to power series inversion of other kinds of matrices is also indicated.