Systems of Equations and the Moore-Penrose Inverse of a Matrix

Abstract

Given a linear algebraic system of m equations in n unknowns written as Ax = b, a standard method to determine the number of solutions is to first reduce the augmented matrix [A,b] to row echelon form. The number of solutions is then characterized by relations among the number of unknowns, rank (A) and rank ([A,b]). In particular, Ax = b is a consistent system of equations, that is, there exists at least one solution, if and only if rank (A) = rank ([A,b]). Moreover, a consistent system of equations Ax = b has a unique solution if and only if rank (A) = n. On the other hand, Ax = b has no exact solution when rank (A) < rank ([A,b]). It is the purpose of this chapter to show how the Moore-Penrose inverse of A can be used to distinguish among these three cases and to provide alternative forms of representations which are frequently employed in each case.