If $A^3=A$ and $\lambda$ is an eigenvalue for $A$, then $\lambda = 0$ or $\lambda = 1$ or $\lambda = -1$

Question

Prove it to be true or false. I really don't have any idea on how to do this. So a little help would be very welcome. Thank you all.

Answer 1

As, $A\vec{x} = \lambda \vec{x}$ ($\vec{x}$ is Eigen vector of $A$)

$A^3\vec{x} = A(A(A\vec{x})) = A(A(\lambda \vec{x})) = \lambda A(\lambda \vec{x}) = \lambda^2 (A\vec{x}) = \lambda^3\vec{x}$

Also,

$A^3 = A$

Thus,

$\lambda \vec{x} = \lambda^3\vec{x}$

$\lambda = \lambda^3$

$\lambda (1- \lambda ^2) = \lambda (1-\lambda )(1+\lambda ) = 0$

Thus,

$\lambda = -1\textrm{ or }0\textrm{ or }1$