If you’re a mathematician, then using Mathematica now and then is unavoidable. However, many people will probably have the same experience as me: I am one of those people who do use Mathematica on a regular basis, but not regular enough to be fluent in it. Here are some concrete case studies showing how I used Mathematica; hopefully, they will show up on Google and be instructive to someone.
Problem
The trace algebra of two matrices is the polynomial ring generated by the traces of products of matrices A and B, like Tr(A), Tr(B), Tr(A.B), Tr(A.A), Tr(A.A.B.A.B). It is known that this trace algebra is the complete set of polynomials that are invariant under simultaneous conjugation of A and B (ie, if one replaces A and B by T^(-1).A.T and T^(-1).B.T, respectively, then the value of the polynomial doesn’t change).