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

Abstract

Let T?=?c 1 T 1?+?c 2 T 2?+?c 3 T 3?-?c 4(T 1 T 2?+?T 3 T 1?+?T 2 T 3), where T 1, T 2, T 3 are three n?x?n tripotent matrices and c 1, c 2, c 3, c 4 are complex numbers with c 1, c 2, c 3 nonzero. In this article, necessary and sufficient conditions for the nonsingularity of such combinations are established and some formulae for the inverses of them are obtained. Some of these results are given in terms of group invertible matrices.