Matrices & Systems · Challenge problem

A $3\times 3$ matrix $A$ satisfies $A^3 = 3I$, and the matrices $I$, $A$, $A^2$ are linearly independent. Then $A + 2I$ is invertible and $(A + 2I)^{-1} = pA^2 + qA + rI$ for unique numbers $p, q, r$. Find $p + q + r$.