More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample.
Let $\mathbb{M}_n$ be the space of $n\times n$ matrices with real entries.
Let $\overline{\mathbb{M}}_n$ be the space of square $n\times n$ matrices with entries $0$ and $1$.
For any $M\in\overline{\mathbb{M}}_n$ we have $$e^M=\exp(M)=\sum_{k=0}^\infty \frac{1}{k!}M^k.$$
Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?
In other words, are there two distinct $M_0,M_1\in \overline{\mathbb{M}}_n$ such that $e^{M_0}=e^{M_1}$?
Since for fixed $n$ the space $\overline{\mathbb{M}}_n$ is finite we can manually check -- it is injective for $n\le4$, but I haven't been able to convince myself for arbitrary $n$.
For $n=4$ there are $2^{n^2}=65536$ matrices to check, and MATLAB's expm and my hastily written matrix building code took half an hour to complete. For $n=5$ we have $2^{25}>3.3\cdot 10^7$ matrices to test which would take ten days to run. For larger $n$ a direct test is out of the question.
The smallest nontrivial examples in my problem are $n=6$, and the interesting ones are $n=12$ and larger.