Let (V, K) be a finite-dimensional vector space and φ ∈ L(V, V ). Find φ such that if A is its matrix representation with respect to some basis of V , then $A^2(I − A) = 0$ but φ is not a projection. This problem shows you that if a map φ satisfies the relation $φ^3 = φ^2$ , this does not necessarily imply that φ is idempotent and hence a projection! Hint: do not attempt to find the most general φ. It is sufficient to provide an example for such a linear transformation in the case $(V, K) = (R^ 2 , R)$.
I've tried to do this by chance and didn't get what satisfies this
Is there a proper way to solve this other than by luck
You want to find some kind of linear map $\varphi$ such that $\varphi^2 \neq \varphi$ but does not change, when you apply it to itself again. Do you know a map $\alpha$ such that for all maps $\beta$ we always have $\alpha \circ \beta = \alpha$? If you have found one, can you find an $\varphi\neq \alpha$ such that $\varphi^2 = \alpha$?