  Let $\mathbb{N}$ be the set of all positive integers and let $a,\, b,\, c$ be
nonzero integers such that $\gcd\left(a,\, b,\, c\right)=1$. In this paper, we
prove the following three results: (1) the solvability of the matrix equation
$aX^m+bY^n=cI,\,X,\,Y\in M_2(\mathbb{Z}),\, m,\, n\in\mathbb{N}$ can be reduced
to the solvability of the corresponding Diophantine equation if $XY\neq YX$ and
the solvability of the equation $ax^m+by^n=c,\, m,\, n\in\mathbb{N}$ in
quadratic fields if $XY=YX$; (2) we determine all non-commutative solutions of
the matrix equation $X^n+Y^n=c^nI,\,X,\,Y\in
M_2(\mathbb{Z}),\,n\in\mathbb{N},\,n\geq3$, and the solvability of this matrix
equation can be reduced to the solvability of the equation $x^n+y^n=c^n,\,
n\in\mathbb{N},\,n\geq3$ in quadratic fields if $XY=YX$; (3) we determine all
solutions of the matrix equation $aX^2+bY^2=cI,\,X,\,Y\in M_2(\mathbb{Z})$.
