Here is the problem statement:
Find a subset $Y$ of $X:=\{A \in \text{Mat}_{2\times 2}(\mathbb{C})\ |\ A^4=A\}$ so that the following occur:
- If $A$ $\in$ $X$ then $\exists$ $B$ $\in$ $Y$ such that $A$ and $B$ are similar.
- If $A$ and $B$ $\in$ $Y$ and $A \ne B$, then they are not similar.
My attempt of solving this problem: If $A$ and $B$ are similar, then this means that they have the same eigenvalues and they also have the same characteristic equation. So essentially we want to find a set, so that within this set every $A$ and $B$ don't have the same eigenvalues, but with every element of $X$ the matrices have the same eigenvalues.
Ok, this is where I get confused, why do we need $A^4=A$?