Question

设A是方阵,A^k=O对某个正整数k成立,求E+A可逆,并求出它的逆

Answer 1

因为 (E+A)(E-A+A^2-.+(-1)^(k-1)A^(k-1))

= E-A+A^2-.+(-1)^(k-1)A^(k-1)

+ A-A^2-.+(-1)^(k-1)A^k

= E +(-1)^(k-1)A^k

= E

所以 E+A 可逆,且 (E+A)^-1 = E-A+A^2-.+(-1)^(k-1)A^(k-1)