Question

设n阶方阵A满足A^2=3A,证明：A-4I可逆,并求出其逆矩阵

Answer 1

由已知,A^2-3A=0

所以 A(A-4E)+(A-4E) +4E = 0

所以 (A+E)(A-4E) = -4E

所以 A-4E 可逆,且 (A-4E)^-1 = -1/4 (A+E).