The reverse order law for \(\{1, 3, 4\}\)-inverse of the product of two matrices (Q961627)

For an \(m \times n\) matrix \(A\) a generalized inverse of \(A\) is an \(n \times m\) matrix \(X\) which satisfies some of the following four Penrose equations: (1) \(AXA=A\), (2)\(XAX=X\), (3) \((AX)^*=AX\), (4) \((XA)^*=XA\). Let \(\emptyset\neq \eta\subseteq \{1, 2, 3, 4\}\). Then \(A\eta\) denotes the set of all matrices \(X\) which satisfy (i) for all \(i \in \eta\). Any matrix \(X \in A\eta\) is called an \(\eta\)-inverse of \(A\). The reverse order law for the generalized inverses has been studied by many mathematicians since the middle 1960s. In this paper, the authors investigate the relationship between \(\{1, 3,4\}\)-inverses of \(AB\) and the product of \(\{1, 3, 4\}\)-inverses of \(A\) and \(B\). They give some necessary and sufficient conditions for \(B\{1, 3, 4\}A\{1, 3, 4\} \subseteq (AB)\{1, 3, 4\}\), \(B\{1, 3, 4\}A\{1, 3, 4\} \supseteq (AB)\{1, 3, 4\}\) and \(B\{1, 3, 4\}A\{1, 3, 4\}=(AB)\{1, 3, 4\}\) and also present two numerical examples.