On nonsingularity and group invertibility of combinations of two group invertible matrices

Abstract

Let $A$ and $B$ be two group invertible matrices, we study the rank,
the invertibility and the group invertibility of $A-B$,
$AA^{\sharp}-BB^{\sharp}$, $c_1A+c_2B$, $c_1A+c_2B+c_3AA^{\sharp}B$
where $c_1, c_2$ are nonzero complex numbers. Under some special
conditions, the necessary and sufficient conditions of
$c_1A+c_2B+c_3AB$ and $c_1A+c_2B+c_3AB+c_4BA$ to be invertible and
group invertible are presented, which generalized some related
results of Ben\'{\i}tez, Liu, Koliha and Zuo [4, 17, 19, 25].