Let $A,M \in \operatorname{Mat}_n(\mathbb{C})$ be $n \times n$ matrices such that $M$ is invertible and $MA \neq AM$. Consider the algebra $\mathcal{A}$ generated by the set $\{I,A,MAM^{-1}\}$, where $I$ denotes the identity. Does there always exist a matrix $T \in \mathcal{A}$ so that $T$ has distinct eigenvalues?
Context : the reason I am interested in this question is because I would like to see precisely how sensitive eigenvalues are with respect to performing really minimal matrix operations.