Elementary Transformations

Q. Suppose $A$ is any $3\times 3$ non-singular matrix and $(A-3I)(A-5I)=O$, where $I=I_3$ and $O=O_3$. If $\alpha A+\beta A^{-1}=4I$, then $\alpha +\beta$ is equal to :

1. $13$
2. $7$
3. $12$
4. $8$