On extensions of Cramer's rule for solutions of restricted linear systems1

Abstract

For the unique solution of a special consistent restricted linear system Ax=bx∊M we derive two different determinantal forms, which both reduce to Cramer's classical rule if A is nonsingular. The representation for the minimum Euclidean-norm solution of Ax=b given recently by Ben-Israel [2] is reobtained in our first approach as special case. A determinantal formula for any {1, 2}-generalized inverse is also given.