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

Abstract

Let T = c1T1 + c2T2 + c3T3 − c4 (T1T2 + T3T1 + T2T3), where T1, T2, T3 are 6 three n × n tripotent matrices and c1, c2, c3, c4 are complex numbers with c1, c2, c3 7 nonzero. In this paper, it is mainly established necessary and sufficient conditions for M n be the set of all n × n complex matrices over C. The symbols rank(A), A * , R(A), 16and N (A) stands for the rank, conjugate transpose, the range space, and the null space of 17A ∈ M n , respectively. Recall that a matrix A ∈ M n is idempotent if A 2 = A and tripotentThe nonsingularity of linear combinations of idempotent matrices and k-potent matrices 26Consider a combination of the formwhere c 1 , c 2 , c 3 ∈ C * , c 4 ∈ C and T 1 , T 2 , T 3 ∈ M n are three tripotent matrices. The 28 purpose of this paper is mainly twofold: first, to establish necessary and sufficient conditions 29 for the nonsingularity of combinations of the form (1.1) and then to give some formulae for 30 the inverse of them. 31Now, let us give the following additional concepts and properties. For a given matrix A ∈ M n is said to be group invertible if there exists a matrix X ∈ M n such thathold. If such an X ∈ M n exists, then it is unique, customarily denoted by A # [3]. A matrix 32A ∈ M n is group invertible if and only if there exist nonsingular S ∈ M n , C ∈ M r such