设A.B为阶方阵,且满足AB=A+B,试证:A-E和B-E均为可逆矩阵

问题描述:

设A.B为阶方阵,且满足AB=A+B,试证:A-E和B-E均为可逆矩阵

AB = A + B
=> AB - A - B = 0
=>A(B - E) - (B-E) = E
=>(A-E)(B-E) = E
=>|A-E| * |B-E| = 1
那么|A-E| 和 |B-E|不等于零
A-E和B-E均为可逆矩阵