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The answer to the question is yes, though I don't have all the old literature at my fingertips. This kind of question for various classes of linear groups has a long history in the study of homomorphisms and isomorphisms of classical groups and then other algebraic groups (van der Waerden, Dieudonne, ...) The most comprehensive treatment was given by Borel and Tits in their Ann. of Math 97 (1973) paper, but emphasizing simple types rather than general reductive groups. Anyway, for general linear groups the ideas occur much earlier and also involve the uniqueness of $n$. (As you point out, the case $n=1$ has a different flavor.) I'll check the sources, but you could also work back from the references in Borel-Tits.