Question about Cayley-Hamilton Theorem

Question

The Cayley-Hamilton theory states that every matrix $A \in M_n(\mathbb{C})$ is a root of its own characteristic polynomial $p_A(x)$. My thought is this,
since by definition $p_A(x) = \det(xI-A)$, filling in $A$ into this polynomial we obtain $$ p_A(A) = \det(AI-A) = \det(A-A) = \det(0) = 0. $$ What is wrong with this reasoning?