# Possible restrictions on generators of $M_n(\mathbb{C})$

Suppose matrices $a$ and $b$ generate $M_n(\mathbb{C})$. I would like to know what restrictions this imposes on $a$ and $b$. More concretely, do there exist $a,b\in M_n(\mathbb{C})$, which generate $M_n(\mathbb{C})$, such that the minimal polynomial of $\alpha a+\beta b$ has degree $\leq n-1$ for all $\alpha,\beta\in \mathbb{C}$?

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