Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs naturally as a conjecture:

For all integers $n,k\geq 2$ there exist three square matrices $A$, $B$ and $C$ of size $k\times k$ and integer entries, such that $\det(ABC)\neq 0$ and: $$A^n + B^n = C^n$$

Of course, the case $k=1$ is just Fermat's Last Theorem, but in that case the conclusion is *the opposite* for $n>2$.

I think that I read somewhere that it is known that the above assertion is true (I do not remember exactly where, and haven't seen anything on Google, but this old question on MSE, on which there is an old reference, that I think does not answer this).

Two observations that are pretty straightforward to verify are the following: the case $2\times 2$ and $3\times 3$ solve the general case $k\times k$ by putting suitable small matrices on the diagonal.

Also, as it is stated in a comment on that question, if the exponent $n$ is odd then the case $2\times 2$ can be solved by this example:

$$\begin{bmatrix} 1 & n^\frac{n-1}{2}\\ 0 & 1\end{bmatrix}^n + \begin{bmatrix} -1 & 0\\n^\frac{n-1}{2} & -1\end{bmatrix}^n = \begin{bmatrix} 0 & n\\1& 0\end{bmatrix}^n$$

Does anybody know of such examples in the $2\times 2$ case for even $n$, and the general $3\times 3$ case?

More clearly: are there easy and *explicit* examples for each $n$ and $k$ for the above conjecture?

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