Take any square matrix and compute its characteristic polynomial. Then substitute the matrix for the variable and start computing. This is a good exercise for a beginner working with matrices. Moreover, the result is quite remarkable. It is, namely, always the zero matrix.

It would require an extraordinary intuition to suspect this, by no means, intuitive result. So this is one of many examples where mathematics surprises us in a most beautiful way. This is the so called Cayley-Hamilton theorem. We hope it will be of interest for a student, who has tried this in some examples, to see a proof of the theorem. Usually the theorem is proved in more advanced textbooks, with techniques not found in a first course of linear algebra. Here we give two elementary proofs, the first rather abstract, the second more concrete.