$A^{3}=A$ $|A^{3}|=|A|$--->(1) for eigen values:Characteristic equation $|A-\lambda I|=0$ $\Rightarrow|A|=|\lambda I|$ $\Rightarrow|A|=|\lambda |$
Cayley–Hamilton Theorem: A square matrix satisfies its own characteristic equation. Put the value in the equation $(1),$ $|\lambda|^{3}=|\lambda|$ $\lambda^{3}-\lambda=0$ $\lambda(\lambda^{2}-1=0$ $\lambda(\lambda+1).(\lambda-1)=0$ so $,\lambda=0,-1,1$
If A^{3}=A then what do you mean by not idempotent matrix???
"Gate is not a 100 m race...