What is wrong with my proof of Cayley-Hamilton? (First year Matrices course)

Question

The characteristic equation of matrix A is p(t)=det(A-tI)

Then $$p(t)=\sum a_rt^r$$

for some coefficients a_r

Then $$p(A)=\sum a_r A^r$$

Let $$\vec{x}$$ be an eigenvector of A with eigenvalue t

But $$p(A) \vec{x} = \sum a_r A^{r} \vec{x} = \sum a_r A^{r-1}t \vec{x} = ... = \sum a_r t^r \vec{x} = det(A-tI) \vec{x} = 0$$

Where have I gone wrong? Much appreciated.