  In this paper we deal with the problem of computing the sum of the $k$-th
powers of all the elements of the matrix ring $\mathbb{M}_d(R)$ with $d>1$ and
$R$ a finite commutative ring. We completely solve the problem in the case
$R=\mathbb{Z}/n\mathbb{Z}$ and give some results that compute the value of this
sum if $R$ is an arbitrary finite commutative ring $R$ for many values of $k$
and $d$. Finally, based on computational evidence and using some technical
results proved in the paper we conjecture that the sum of the $k$-th powers of
all the elements of the matrix ring $\mathbb{M}_d(R)$ is always $0$ unless
$d=2$, $\textrm{card}(R) \equiv 2 \pmod 4$, $1<k\equiv -1,0,1 \pmod 6$ and the
only element $e\in R \setminus \{0\}$ such that $2e =0$ is idempotent, in which
case the sum is $\textrm{diag}(e,e)$.
