Show whether $\lambda$ an eigenvalue of $E$ is an element of ${\pm{1}}$ when $E^{2}$ = $I$

Question

When $E^{2} = I$ (where $I = n×n$ identity matrix), show that if $\lambda$ is an eigenvalue of $E$ then λ $\in$ {$\pm1$}.

Answer 1

$E^{2} = I$ implies that $E^{2} x=Ix=x$. So we have to solve $E=ax$ such that $a^2=1$ since

$E^{2} x=Ix=x=EEx=Eax=aax$. And the only values for $a$ are one and negative one

($a^2=1$). IF $a$ is an eigenvalue of E

Answer 2

If $x\neq 0$ and $Ex=\lambda x$ then $x=E^2x=\lambda^2 x$, so $(\lambda^2-1)x=0$.