Matrix inversion by a Monte Carlo method

The following unusual method of inverting a class of matrices was devised by J. von Neumann and S. M. Ulam. Since it appears not to be in print, an exposition may be of interest to readers of MTAC. The method is remarkable in that it can be used to invert a class of re-th order matrices (see final paragraph) with only re2 arithmetic operations in addition to the scanning and discriminating required to play the solitaire game. The method therefore appears best suited to a human computer with a table of random digits and no calculating machine. Moreover, the method lends itself fairly well to obtaining a single element of the inverse matrix without determining the rest of the matrix. The term "Monte Carlo" refers to mathematical sampling procedures used to approximate a theoretical distribution [see MTAC, v. 3, p. 546]. Let B be a matrix of order re whose inverse is desired, and let A = I — B, where / is the unit matrix. For any matrix M, let \T(M) denote the r-th proper value of M, and let M,-,denote the element of M in the i-th row and j-th column. The present method presupposes that