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)=x*^{2}-3x+2. What the Cayley-Hamilton

theorem
says is that

*A*^{2}-3A+2I is the zero matrix. Try it!