Adjoint Mappings and Inverses of Matrices

Abstract

In this paper, we investigate a general and determinantal representation, and conditions for the existence of a nonzero {2}-inverse X of a given complex matrix A. We introduce a determinantal formula for X, representing its elements in terms of minors of order s = rank (X), 1 ≤ s ≤ r = rank (A), taken from the matrix A and two adequately selected matrices. In accordance with these results, we find restrictions of the adjoint mapping such that the set A{2} is equal to the union of their images. Minors of {2}-inverses are also investigated. Restrictions to the set of {1, 2}-inverses produce the known results from [1–3, 10]. Also, in a partial case, we get known results from [11] relative to the Drazin inverse.