  In this paper, we shall be considering the Waring's problem for matrices. One
version of the problem involves writing an $n \times n$ matrix over a
commutative ring $R$ with unity as a sum of $k$-th powers of matrices over $R.$
This study is motivated by the interesting results of Carlitz, Newman,
Vaserstein, Griffin, Krusemeyer, Richman etc. obtained earlier in this
direction. The results are for the case $n \geq k \geq 2$ in terms of the trace
of the matrix. For $n < k,$ it was shown by Katre, Garge that it is enough to
work with the special case $n = 2$ and $k \geq 3.$ The cases $3 \leq k \leq 5$
and $k = 7$ were settled in earlier results. There was no case of a composite,
non-prime-power $k$ occuring above. In this paper, we will find the trace
criteria for a matrix to be a sum of sixth (a composite non-prime power) and
eighth powers of matrices over a commutative ring $R$ with unity. An elegant
discriminant criterion was obtained by Katre and Khule earlier in the special
case of an order in an algebraic number field $\mathcal{O}.$ We will derive
here similar discriminant criteria for every matrix over $\mathcal{O}$ to be a
sum of sixth and eighth powers of matrices over $\mathcal{O}.$
