Cayley-Hamilton Theorem for matrices

The Cayley Hamilton Theorem states that if $\displaystyle T$ is a linear operation on vector space $\displaystyle V$ and $\displaystyle f(t)$ is the characteristic polynomial of $\displaystyle T$, then $\displaystyle f(T)$ is the zero transformation.

How do I extend this to matrices? As in, how do I show that: if $\displaystyle A$ is $\displaystyle n \times n$ and $\displaystyle f(t)$ is the characteristic of $\displaystyle A$, then $\displaystyle f(A)$ is the zero matrix?

I attempted to prove it using $\displaystyle f(A) = det(A - AI) = det(O) = 0$ but this reasoning seems faulty...