If $A$ is an $n \times n$ matrix such that $A^3 = O_{3}$, show that $I - A$ is invertible with inverse $I + A + A^2$

Question

So this question is basically a proof.

If $A$ is an $n \times n$ matrix (so square) which satisfies the condition $A^3 = O_{3}$ ($A^{3}$ gives the $3 \times 3$ zero matrix), then show that $(I - A)$ is invertible with inverse $(I + A + A^2)$.

I have no idea where to start, all help welcome.

Answer 1

Hint: for any invertible matrix $A$ with inverse $A^{-1}$, we have $AA^{-1} = I$. Try multiplying out $(I-A)(I+A+A^{2})$. You should get some cancellation and then use your condition to conclude something...