Fermat's and Catalan's equations over $M_2(\mathbb{Z})$

Hongjian Li, Pingzhi Yuan

Let $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in M_2\left(\mathbb{Z}\right)$ be a given matrix such that $bc\ne 0$ and let $C(A)=\{B\in M_2(\mathbb{Z}):AB=BA\}$. In this pap…