So here is a question I have been working on, any suggestions would be great.

If $\displaystyle B$ is an $\displaystyle n x n$ matrix with enteries in $\displaystyle F$, then prove that there is a nonzero polynomial $\displaystyle p \in F[t]$ which has $\displaystyle B$ as a root.

So here is what I have so far. There must exist $\displaystyle a_0,..., a_r$ not all 0 such that

$\displaystyle p(t) = a_0 + a_1t + ... + a_rt^r$

$\displaystyle p(B) = a_0I + a_1B + ... + a_rB^r = 0$

So pretty much from this point we just need to show that the set $\displaystyle I,B,B^2,...,B^r $ is linearly dependent. Any advice on how to go about doing this?