Use the Cayley-Hamilton Theorem!
So here is a question I have been working on, any suggestions would be great.
If is an matrix with enteries in , then prove that there is a nonzero polynomial which has as a root.
So here is what I have so far. There must exist not all 0 such that
So pretty much from this point we just need to show that the set is linearly dependent. Any advice on how to go about doing this?
Use the Cayley-Hamilton Theorem!
Thank you both very much for your help, however I am still a bit confused. We have not studied determinants (except briefly for 2x2 matrices). Would it be possible for you to give a short explanation of your proof?
The wikipedia page dealt a lot with determinants, eigenvectors and eigenvalues, all of which we have not yet been taught, so I found the proofs hard to follow.
Thanks!