| Let A be an nxn matrix over a field k
(think of k as the real numbers or complex numbers).
The characteristic polynomial of A is c(x)=det(xI-A)
and it is a familiar fact that the zeroes of this polynomial are the
eigenvalues of A.
Much more remarkable is:
Cayley-Hamilton Theorem The matrix A satisfies its own
characteristic equation. That is c(A)=0.
It's worth looking at an example to understand what this result actually means.
Take A=
-- --
| 1 1 |
| 0 2 |
-- --
Then c(x)=x2-3x+2. What the Cayley-Hamilton theorem
says is that A2-3A+2I is the zero matrix. Try it! | 
