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

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.

• If the minimal polynomial of the matrix is $X-1$, it can. But this means the matrix is the identity matrix. Jul 7 '20 at 14:54

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.