Determinantal representation of \(\{i,j,k\}\) inverses and solution to linear systems (Q2702796)

For an \(m\times n\) complex matrix \(A\) the following equations have been considered: (1) \(AXA=A\), (2)\(XAX=X\), (3) \((AX)^*=AX\), (4) \((XA)^*=XA\) and (if \(m=n\)) (5) \(AX=XA\). A matrix \(X\) is called \(\mathcal S\)-inverse of \(A\) if it satisfies conditions contained in a subset \(\mathcal S\subset \{1,2,3,4,5\}\). NEWLINENEWLINENEWLINEThe most important result of the paper (Theorem~2.1) gives a determinantal characterization of \(\{1,2\}\)-inverse. Then it is applied to determinantal representation of \(\{1,2,3\}\) and \(\{1,2,4\}\) inverses and used to obtain a determinantal representation of the solution of a system of linear equations. The obtained results generalize some previous ones of the author and \textit{M. Stanković} [Mat. Vesnik 46, 41-50 (1994; Zbl 0840.15005)].