整数环上一类二阶矩阵方程的解

The Solution of 2×2 Matrices Equations A^n=kE with Integer Entries

设A是一个m×m可逆矩阵,称使得An=kE(E为单位矩阵)对某个实数k成立的最小正整数n为A的阶,记为O(A).本文证明,在整数环上,2×2矩阵方程An=kE(det(A)≠0)有解当且仅当矩阵A的阶O(A)∈{1,2,3,4,6}.

A matrix is said to have order n （possibly infinite） if m is the smallest positive integer with A^n=kE for some k∈R. In this paper, the author prove that the solution of the system of equations （a c b d ）=k（1 0 0 1） with a,b,c, d∈ Z and ad-bc≠0 exists if and only if（a c b d ） has the order 1,2,3,4, or 6.