On g-inverses and the nonsingularity of a bordered matrix \(\begin{pmatrix} A&B\\ C&0\end{pmatrix}\)
The matrix \(M=\begin{pmatrix} A & B \\ C & 0\end{pmatrix}\) satisfies the rank additivity conditions if \(\text{rank}M = \text{rank}(A,B)+ \text{rank}(C,0)= \text{rank}(A^ T,C^ T)+ \text{rank}(B^ T,0).\) Let \(G=\begin{pmatrix} C_ 1 & C_ 4 \\ C_ 3 & -C_ 4\end{pmatrix}\) be a g-inverse of M. Necessary and sufficient conditions are given for \(C_ 1\) to be a g-inverse, outer inverse, reflexive inverse, and Moore-Penrose inverse of A. Necessary and sufficient conditions are also given for M to be nonsingular along with explicit expressions for the \(C_ i\) in \(M^{-1}\).
scientific article; zbMATH DE number 4146162