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

Abstract

Let T 1 and T 2 be two n × n tripotent matrices and c 1 , c 2 two nonzero complex numbers. We mainly study the nonsingularity of combinations T = c 1 T 1 + c 2 T 2 − c 3 T 1 T 2 of two tripotent matrices T 1 and T 2 , and give some formulae for the inverse of c 1 T 1 + c 2 T 2 − c 3 T 1 T 2 under some conditions. Some of these results are given in terms of group invertible matrices.

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

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Let A and B be two group invertible matrices, we study the rank, the nonsingularity and thewhere c 1 , c 2 are nonzero complex numbers. Under some special conditions, the necessary and sufficient conditions of c 1 A + c 2 B + c 3 AB and c 1 A + c 2 B + c 3 AB + c 4 BA to be nonsingular and group invertible are presented, which generalized some related results of Benítez, Liu, Koliha and Zuo [4,17,19,25].

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“…So Theorems 2.12,2.13 are the generalizations of the two Theorems of [4]. Theorem 2.14 is the generalization of Theorem 2.6 of reference [19].…”

“…In [2,[5][6][9][10][11][13][14][16][17]20,22,25], the authors described the nonsingularity, Drazin invertibility, group invertibility of the linear combinations of two idempotent matrices. In particular, the authors studied the nonsingularity and group invertibility of linear combinations of two k-potent matrices and two group invertible matrices in [4,19]. The invertibility of combinations of k-potent operators and commuting generalized and hypergeneralized projectors was considered in [12] and [21].…”