Cayley Hamilton Theorem Question

Hello, a tad stuck on an algebra assignment.

The Cayley Hamilton theorem implies that, for any nxn matrix A over a field K, there is a polynomial p(x) with coefficents in K and degree n in x such that p(A)=0.

By considering the matrices A^i for ,0 less than or eqaul to (i) less than or equal to (n^2), prove, without using the cayley hamilton theorem, that there is a non-zero polynomial p(x) of degree at most (n^2) in x, such that p(A)=0.

Hints/pointers/answers will all be welcomed.

Thankyou