Prove that $A$ in invertible through characteristic polynomial: $x^{500}+x^{100}-x+4$

Question

Let $A$ be a matrix with charecteristic polynomial $$p(x)=x^{500}+x^{100}-x+4$$ Prove that $A$ is invertible.

I'm very lost with this one, because I don't know how to calculate the eigenvalues, I thought that I maybe have to prove that the polynomial has differents eigenvalues, but I'm not sure.

Answer 1

A matrix is invertible if and only if the determinant is nonzero, which is true if and only if the constant term of the characteristic polynomial is nonzero.

Answer 2

By the Cayley–Hamilton theorem,$$A^{500}+A^{100}-A+4\operatorname{Id}=0;$$in other words,$$A.(A^{499}+A^{99}-\operatorname{Id})=-4\operatorname{Id}.$$Can you take it from here?

Answer 3

We have $p(0)=4 \ne 0$, hence $0$ is not an eigenvalue of $A$. Therefore $A$ is invertible.