Can an $n \times n$ matrix satisfy an $n$ degree polynomial equation other than its characteristic polynomial equation?

Question

Can an $n \times n$ matrix satisfy an $n$ degree polynomial equation other than its characteristic polynomial equation?

I was curious if the characteristic polynomial equation is the only $n$ degree equation that can be satisfied by a matrix. I have tried by trial and error to make up an equation for $2\times 2$ matrix but always end up with the characteristic polynomial.

Answer 1

If the eigenvalues are distinct, then the characteristic polynomial is the only monic $n$th degree polynomial the matrix satisfies, since the eigenvalues are the roots, and so determine an $n$th degree polynomial up to a constant factor. If the minimal polynomial is the characteristic polynomial, we can make a similar statement. If not, then it's not true as ancientmathematician's answer shows.

Answer 2

The $2\times 2$ identity matrix satisfies $X(X-1)$.

Answer 3

The question (supposing you implicitly don't want to talk about scalar multiples of the characteristic equation) is whether there exist, in the degree $n$ of the characteristic polynomial, any polynomial multiples of the minimal polynomial (i.e., annihilating polynomials) that are not (necessarily scalar) multiples of the characteristic polynomial. It is easy to see that the reply is: such multiples exists if and only if the minimal polynomial is of degree strictly less than $n$ (just multiply it by a factor of the necessary degree that is not a scalar multiple of the exact quotient of the characteristic polynomial by the minimal polynomial; this is clearly possible once that quotient is not the constant polynomial$~1$. Examples are easily found; particularly easy ones involve scalar multiples of the identity matrix, whose minimal polynomial is of degree$~1$.