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

Abstract: Let $A=\begin{pmatrix} a & b \ c & d \end{pmatrix}\in M_2\left(\mathbb{Z}\right)$ be a given matrix such that $bc\neq0$ and let $C(A)={B\in M_2(\mathbb{Z}): AB=BA}$. In this paper, we give a necessary and sufficient condition for the solvability of the matrix equation $uX^i+vY^j=wZ^k,\, i,\, j,\, k\in\mathbb{N},\, X, \,Y,\, Z\in C(A)$, where $u,\, v,\, w$ are given nonzero integers such that $\gcd\left(u,\, v,\, w\right)=1$. From this, we get a necessary and sufficient condition for the solvability of the Fermat's matrix equation in $C(A)$. Moreover, we show that the solvability of the Catalan's matrix equation in $M_2\left(\mathbb{Z}\right)$ can be reduced to the solvability of the Catalan's matrix equation in $C(A)$, and finally to the solvability of the Catalan's equation in quadratic fields.