I have a exercise in my linear algebra textbook:
Let $c_2\lambda^2+c_1\lambda +c_0=0$ be the characteristic equation for the matrix $$A=\begin{pmatrix}1&3\\3&1\end{pmatrix}$$
Prove that $c_2A^2+c_1A +c_0I=0$
This is Cayley-Hamilton theorem.
My solution:
If a root $\lambda$ exists, then it is the eigen value for the eigen vector $\vec{v}$. If we multiply $c_2A^2+c_1A +c_0I$ with $\vec{v}$ then we get: $$(c_2\lambda^2+c_1\lambda +c_0)\vec{v}$$ and since the eigen vector is not the zero vector and $c_2\lambda^2+c_1\lambda +c_0=0$ is true, $c_2A^2+c_1A +c_0I=0$ is also true.
Is this enough to prove the theorem and solve the problem?