<s>[INST] Matrix Equation $B^3+C^3=\begin{pmatrix}-1 & 1\\ 0 & -2\end{pmatrix}$ How can I solve in $\mathcal{M}_{2}(\mathbb{Z})$ the equation $$B^3+C^3=\begin{pmatrix}-1 & 1\\ 0 & -2\end{pmatrix}?$$
I try to use $$X^2-Tr(X)X+det X\cdot I_2=0_2$$ but there are $B$ and $C$, I don't still obtain anything. 
thanks. 
 [/INST] The easiest part. We assume that $B,C$ are upper-triangular.
Then $b_{1,1}^3+c_{1,1}^3=-1$; according to Fermat (up to order) $b_{1,1}=-1,c_{1,1}=0$.
And $b_{2,2}^3+c_{2,2}^3=-2$, that implies $b_{2,2}=c_{2,2}=-1$ (see $(a+1)^3-a^3$ when $a>0$).
By identification, we conclude that there is an integer $p$ s.t. (up to order)
$B=\begin{pmatrix}-1&p\\0&-1\end{pmatrix},C=\begin{pmatrix}0&1-3p\\0&-1\end{pmatrix}$.
EDIT. There are solutions $(B,C)$ in $M_2(\mathbb{R})$ s.t. $B^3,C^3$ are not upper-triangular; for example
$B\approx \begin{pmatrix}0&-2.0107\\1&-1.8503\end{pmatrix},C\approx\begin{pmatrix}1.1426&-2.7184\\1&0\end{pmatrix}$.
Thus the question is: can we find such matrices in $M_2(\mathbb{Z})$ ?
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