"Given operators σ,τ on a finite-dimensional space V, show that στ=i, and that σ=p(τ) for some polynomial p in F[x]." The first part was no problem. As for the second, I have a strong suspicion that p is the characteristic polynomial, mostly because I believe I heard of that fact before (that a matrix inserted into its own characteristic polynomial it its inverse). However, I can't seem to find anything about that, and furthermore, the characteristic polynomial has not yet been mentioned in the text I'm using. Any idea how the proof should proceed?