若n阶矩阵A满足A的3次幂等于3A(A-I),试证I-A可逆,并求（I-A)的-1次幂

题目

若n阶矩阵A满足A的3次幂等于3A(A-I),试证I-A可逆,并求（I-A)的-1次幂

答案

由 A^3 = 3A(A-I)
得 (A^2-2A+ I)(I-A) = I
所以 I-A 可逆,且
(I-A)^(-1) = A^2-2A+ I
(A-I) 因子:
由 A^3 = 3A(A-I)
得:A^3 -3A^2 +3A = 0
A^2(A-I) +A^2 - 3A^2 +3A = 0
A^2(A-I) -2A(A-I) -2A +3A = 0
A^2(A-I) -2A(A-I) + (A-I) + I = 0
(A^2-2A+ I)(A-I) = -I
(A^2-2A+ I)(I-A) = I