The result mentioned above is Waring's problem for matrices (respectively for algebraic
number fields). The result is:

Theorem (Katre, Kuhle 1990): Let $R$ be an order in an algebraic number field $K$.
Let $n\ge k \ge 2$. Then every $n\times n$ matrix over $R$ is a sum of $k$-th powers
if and only if it is the sum of *seven* $k$-th powers if and only if $(k, disc (R)) = 1$.

If this is true, then the question is if the seven matrices can be constructed explicitly.
For the case $k=n=2$ and $R=\mathbb{Z}$ this seems to be the case (article of Newman
"Sums of squares of matrices"), by constructing certain companion matrices to characteristic polynomials: every integral
$2 \times 2$ matrix is the sum of at most $3$ integral squares. The proof for the general
case however uses an argument of the form "every $n\times n$-matrix is a sum of $k$-th powers
in $M_n(R)$ if every $m\times m$-matrix is a sum of $k$-th powers in $M_m(R)$ for
$n\ge m\ge 1$ and $k\ge 2$", which does not seem to be constructive.