Ljunggren's trinomials and matrix equation $A^{x}+A^{y}=A^{z}$

Abstract

We give some necessary and sufficient conditions for solvability of the matrix equation (*) $A^x + A^{y}=A^{z}$, with certain restrictions on integers $x, y, z$ and a matrix $A \in M_{k}(\bm{Z})$, by applying Ljunggen's result on trinomials. Moreover, we obtain full solution of (*) for the case $k=2$ by another technique.