A Class of Methods for Inverting Matrices

The present note is perhaps of theoretical interest only but is intended to exhibit in somewhat general perspective the classical elimination methods for inverting matrices, and a seemingly different one proposed recently by Hestenes (1953, 1957). These will be seen to be special cases of a general class of possible methods, although possible new variants may or may not turn out to possess practical advantages. The description of the class as a whole will be based upon the following: LEMMA. Let P be a nonsingular matrix of order n, and let Q be a matrix of rank n r at most. Then if P Q has rank r + 1 at least, there exist column vectors u and v and a scalar ar such that I o-uv is nonsingular and