For a $2 \times 2$ matrix having eigenvalues 1,1 will the matrix satisfy a two degree monic polynomial other than characteristic polynomial?

Question

For a 2 X 2 matrix (except the identity matrix) having eigenvalues 1,1 is it necessary for the matrix to satisfy a two degree monic polynomial (X-1)(X-K) for some real K (K is not equal to 1) (the matrix clearly cannot satisfy a monomial).
For example, an identity matrix can satisfy X(X-1) (the identity matrix also has eigenvalues 1,1) but can we always find an n degree polynomial except the characteristic polynomial for all n x n matrices having a repeated eigenvalue.

Answer 1

The matrix must satisfy its characteristic polynomial, i.e. $(X-I)^2=0$. If also $(X-I)(X-kI) = 0$, then their difference $(X-I)^2 - (X-I)(X-kI) = (X-I)(k-1) = 0$ so if $k \ne 1$, $X-I = 0$ i.e. $X$ is the identity matrix.