Matrices over commutative rings as sums of fifth and seventh powers of matrices

Certain trace conditions were obtained by Katre-Garge for writing matrices over a commutative ring R with unity as a sum of kth powers of matrices over R. In particular, they derived a discriminant criterion for cubes and fourth powers of matrices, using the above trace conditions for the ring of integers as well as orders in algebraic number fields. In this paper, we use the above trace criterion to obtain similar results for fifth and seventh powers of matrices, and to get an analogous discriminant criterion for rings of integers of algebraic number fields and orders therein. The proof presented here explicitly constructs the matrices involved in the calculations.