TITLE: Does the Cayley–Hamilton theorem work in the opposite direction?
QUESTION [9 upvotes]: The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic equation.
But does it work in the opposite direction?
If for example for a certain matrix $A$ we know that
$ A^2-6A+9I=0, $
does that mean that the characteristic equation of $A$ is
$
\lambda^2-6\lambda+9=0
$ 
?

REPLY [6 votes]: No. In general if a matrix is a root of a polynomial, that polynomial is a multiple  of the minimal polynomial of that matrix. 
Even if you only consider n-degree polynomials for an n×n matrix, this still can fail. 
For example, if A=0, then the characteristic polynomial is $x^n$, but A is the root of any polynomial $xg(x)$.
However, all is not lost. If your minimal polynomial equals your characteristic polynomial (as with companion matrices or in the case you have no repeated roots), then the characteristic polynomial (= minimal polynomial) will be the unique monic n-degree polynomial with root A.