The nullity and rank of combinations of two outer inverses of a given matrix (Q987943)

This paper shows that the nullity and rank of \( aP +bQ-cQAP \) is a constant, where \(P\) and \(Q\) are outer inverses of a given matrix \(A\), \(c = a + b\) \((a, b \neq 0) \) or \(c \neq a + b\), \(a, b, c\in C\). In addition, the rank of \( aP +bQ-cQAP \) is equal to the rank of \(P -Q\) if \(c = a+b\) and to that of \(P + Q\) if \(c \neq a + b\).