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This talk is based on the paper with the same title which appeared in Electronic Journal of Linear Algebra, November 2021 it is dedicated to the memory of dear Professor Cem Tezer.

A new necessary and sufficient condition for the existence of an m-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries. Thus, computation of the Jordan form of the m-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an m-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. For a singular matrix having an m-th root with a pair of nilpotent Jordan blocks of sizes s and l, a new m-th root is constructed by replacing that pair by another one of sizes s + i and l − i, for special s, l, i. If time permits we can state some results for the existence of m-th roots of A^k for a matrix A over an arbitrary field that is a sum of two commuting matrices where k ≥ t and t is the nilpotency of the nilpotent part of A.