The inverse of this type of matrix is required in various areas of application of statistical, stochastic control and communication theory. Examples range over i) the fitting of autoregressive series (Siddiqui, 1958; Grenander and Rosenblatt, 1957), ii) the fitting of regression functions when the errors are serially correlated (Hannan, 1960; Whittle, 1963), iii) the estimation of shaping and matching filters in communication theory (Robinson, 1966; Wiener, 1949), iv) adaptive estimation of control system parameters by Bayesian methods (Aoki, 1967). Whenever any of the above applications require large order matrix inversion of a Toeplitz matrix and require it dynamically for the purpose of updating parameters, then it is important to reduce the amount of computation wherever feasible. It is the purpose of this note to give new matrix forms for the inverse of a finite Toeplitz matrix which produce a definite saving in computation.