TITLE: Generalization to higher dimension of $e^r \not \in \mathbb Q$
QUESTION [6 upvotes]: The following is well known and not difficult to prove: 
$$\forall r \in \mathbb Q^*,  e^r \not \in \mathbb Q.$$
See for instance https://proofwiki.org/wiki/Exponential_of_Rational_Number_is_Irrational 
Could this be generalized with the following result, for $n\geq 2$:
$$\forall M \in \mathrm{GL}(n,\mathbb Q),  \exp(M) \in \mathrm{GL}(n,\mathbb R)\setminus \mathrm{GL}(n,\mathbb Q)?$$
If yes, do you have any proof or reference?

REPLY [2 votes]: Actually, a stronger result is true: If $A$ is a non-singular matrix with algebraic (over $\mathbb{Q}$) entries, then $\exp A$ has at least one transcendental entry. To prove this, you can use the Jordan canonical form and assume that $A$ is upper triangular. Since $A$ is non-singular, we have $A_{ii}\neq 0$ for some $i$, and so the $(i,i)$ entry of $\exp A$ is transcendental by Lindemann--Weierstrass.
(The above argument turned out to be contained essentially in the earlier answer by Pierre-Guy.)