### On the Sum of Strictly *k*-zero Matrices

#### Abstract

Let *k* be an integer such that *k* ≥ 2. An* n*-by-*n* matrix* A* is said to be strictly* k*-zero if *Ak* = 0 and *Am* ≠ 0 for all positive integers *m* with *m* < *k*. Suppose* A* is an *n*-by-*n* matrix over a field with at least three elements. We show that, if* A* is a nonscalar matrix with zero trace, then (i) *A* is a sum of four strictly *k*-zero matrices for all *k* ∈{2,..., n}; and (ii) *A* is a sum of three strictly *k*-zero matrices for some *k* ∈{2,..., *n*}. We prove that, if *A* is a scalar matrix with zero trace, then* A* is a sum of five strictly *k*-zero matrices for all *k* ∈{2,..., *n*}. We also determine the least positive integer *m*, such that every square complex matrix* A* with zero trace is a sum of *m* strictly *k*-zero matrices for all *k* ∈{2,..., *n*}.

* Keywords:* Nilpotent matrix, trace, Jordan canonical form