Explicit expressions of the generalized inverses and condensed Cramer rules

In this paper, we obtain an explicit representation of the {2}-inverse AT,S(2) of a matrix A ∈ Cm×n with the prescribed range T and null space S. As special cases, new expressions for the Moore-Penrose inverse A+ and Drazin inverse AD 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 ∈ R(Ak), b ∈ R(Ak), k = Ind(A). Finally, condensed determinantal expressions for A+, AD, AA+, A+A, and AAD are also presented.