Explicit expressions of the generalized inverses and condensed Cramer rules

In this paper, we obtain an explicit representation of the {2}-inverse A(T, S)((2)) of a matrix A is an element of C-m x n with the prescribed range T and null space S. As special cases, new expressions for the Moore-Penrose inverse A(+) and Drazin inverse A(D) are derived. Through explicit expressions, we re-derive the condensed Cramer rules of Werner for minimal-norm least squares solution of linear equations Ax = b and propose two new condensed Cramer rules for the unique solution of a class of singular system Ax = b, x is an element of R(A(k)), b is an element of R(A(k)), k = Ind(A). Finally, condensed determinantal expressions for A(+), A(D), AA(+), A(+)A, and AA(D) are also presented. (c) 2005 Elsevier Inc. All rights reserved.