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!