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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:

  1. If $A$ $\in$ $X$ then $\exists$ $B$ $\in$ $Y$ such that $A$ and $B$ are similar.
  2. 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 $A^4=A$?

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