Is my reasoning here correct?

Question

Let's say that we have a matrix $A \in M_n{\mathbb{C}}$ such that $A^2=A-I_n$.
Now,I want to see what I can say about its eigenvalues. I think that they are roots of the polynomial equation $x^2-x+1=0$(of course,they may come in any multiplicity).
Is it true that all of $A$'s eigenvalues need to be roots of this equation(and have different multiplicities) or do only some of them need to be and the others can be any complex numbers? I believe that they all are roots of this equation and have different multiplicities,but I am not so sure.

Answer 1

If $Ax=\lambda x$ with $x \neq 0$ then $A^{2}x=(A-I_n)x$ gives $\lambda^{2}-\lambda+1=0$ so $\lambda$ is necessarily a root of $\lambda^{2}-\lambda+1=0$.

PS: no, knowledge of Caley Hamilton Theorem etc are needed for this.