On some properties of sequences of traces of powers of matrices

In this paper, we study sequences of the form $ \{\mathrm{tr}(A^k)\}_{k \in \mathbb {N}}, $ where $ A $ is a matrix with components from some field. For such sequences, the conditions for their periodicity are found, and the results from the works by A.M. Bikchentaev, P.N. Ivanshin (2021) and by S.O. Shatunovsky (1903) are obtained as corollaries.