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If A epsilon M_n (R) with A^3 = 0, show I - A is nonsingular with (I - A)^-1 = I + A + A^2.

(I-A)*(I + A+A^2) = I + A + A^2 -(A +A^2 +A^3) = I +A +A^2 - A - A^2 -A^3 = I - A^3 = I

=>
(I-A)*(I + A+A^2) = I...........................(1)

(I + A+A^2)(I-A) = I + A + A^2 - A -A^2 -A^3 = I- A^3 = I

=>
(I + A+A^2)(I-A) = I...........................(2)

From (1), (2), we can conclude that

(I-A)^-1 = I + A + A^2, which also imply that I-A is non singular ( since inverse exists for that matrix)

thus proved