How to construct a matrix $4×4$ with two parameters

Question

I have been searching for a method to construct a $4×4$, $5×5$ or $6×6$ matrix $A$ with two parameters $α$ and $β$ satisfying this property:

1. If $α = β = 0$, then $A$ has no eigenvalue equal to $0$.
2. If at least one of $α$ and $β$ equal to one, then $A$ has an eigenvalue equal to $0$.

Is there any known method to construct such $A$? Any help is appreciated.