Abelianization of GL_n

For any field $k$, the abelianization of $GL_n(k)$ is $k^{\ast}$ except for $n=2$ and $k=\mathbb{F}_2$ or $\mathbb{F}_3$. I think this is in Lang's Algebra, Chapter XIII sections 8 and 9. (I say I think because I am relying on google books here, and some key pages are missing.)

EDIT: I just looked back at this old answer, and the abelianization of $GL_2(\mathbb{F}_3)$ is $\mathbb{F}_3^{\times}$ as well, so $GL_2(\mathbb{F}_2)$ is the only counterexample. I think that what I was thinking when I wrote this is that the abelianization of $SL_2(\mathbb{F}_3)$ in nontrivial.