Nonsingularity from an annihilating polynomial

Suppose $f$ is a polynomial of degree $n$ with real coefficients, and $A$ is an $n\times n$ matrix with real entries satisfying $f(A)=0$. Consider the following statements: (I) If $f(0)\neq 0$, then $A$ is necessarily nonsingular. (II) If $f(0)=0$, then $A$ is necessarily singular. Then