I can´t resist to mention the Cayley-Hamilton theorem. Something intuitively correct turns out to be mathematically correct too, but for non-intuitive reasons! I still remember, its proof (I´m here referring to the one using the correspondence between operation and representation) worked from my perspective like a magic, clear, simple, non-trivial and beautiful, and it also made me interested in algebra, beyond the lecture in linear algebra for first-year students. It was nice time...