TITLE: A Fermat's FLT look-like for matrices.
QUESTION [8 upvotes]: I was wonderwing about a matrix equation (in some way similar to a very well known theorem :P).
Find all positive integers $n,m$ such that there exists $X$, $Y$, $Z$ matrices $n\times n$, non-singular, such that: $$X^m+Y^m=Z^m$$
Disclaimer: $X$, $Y$, $Z$ must have integer entries. Also, they may have an inverse which is not necessarily in $\mathbb{Z}^{n\times n}$.
I made some approachs (I don't know how to spoil, so if you want to think it yourself stop reading):
As my intuition says to me that there will always be solutions for $n>1$, then, it suffices to show there will be solutions for $n=2$ and $n=3$ (because then you can form a matrix composed by matrices $2\times 2$ and $3\times 3$ in its diagonal).
I showed that for all $m$ odd, and $n=2$ there is a solution. 
Can someone give me a hand with the other cases? I would be very grateful.

REPLY [9 votes]: Ribenboim lists some known results in his book 13 Lectures on Fermat's Last Theorem, e.g..