What is the reason for so much big prof of Cayley-Hamilton theorem in Linear Algebra?

Question

As I was studying linear algebra from various books I came to know about Cayley -Hamilton theorem. It states that: An $n×n$ matrix satisfies it's own characteristic equation. I see in various books that they have given a big proof for this. But this seems quite obvious to me. Because characteristic equation of an $n×n$ matrix A is given by: $det(xI_n-A)=0$, where $I_n$ is the $n×n$ identity matrix. If we put $x=A$ in the lhs of the characteristic equation we get: $det(A.I_n-A)=det(A-A)=det(O_n)=0$, where $O_n$ is the zero matrix of order n. Is there any mistake in my idea? If not then why are they giving so much bigger proofs.

Answer 1

I just discovered the answer from a similar post in stack: You can't just replace $x$ by $A$ since the operation between $x$ and $I_n$ in $det(xI_n-A)$ is the scalar multiplication. But when you put $x=A$ the operation becomes matrix multiplication.