Let $A$ be a square matrix such that $A^3 = 2I$

i) Prove that $A - I$ is invertible and find its inverse

ii) Prove that $A + 2I$ is invertible and find its inverse

iii) Using (i) and (ii) or otherwise, prove that $A^2 - 2A + 2I$ is invertible and find its inverse as a polynomial in $A$ 

$I$ refers to identity matrix. 

Am already stucked at part i). Was going along the line of showing that $(A-I)([...]) = I$ by manipulating the equation to $A^3 - I = I$ and I got stuck... :(