Number of solutions of a matrix equation(TIFR GS 2011).

Question

Let $A$ be a $2\times 2$ matrix with complex entries,The number of $2\times 2$ matrices with complex entries such that $A^3=A$ is infinite.

Proof: Let us consider matrices of the form,$A=\begin{bmatrix} 1 & z\\ 0 & -1 \\ \end{bmatrix}$,where $z\in \mathbb C$.,this is an infinite collection each matrix having characteristic equation $x^2-1=0$.By Cayley-Hamilton theorem,each of these infinitely many matrices satisfy, $A^2-I=O$.So,$A^3=A$ is satisfied by all of these matrices,although I am not saying that this collection is complete.

Can someone help me find the complete collection?

Answer 1

HINT

The eigenvalues can only be $0,1,$ or $-1$, so one can write down all possible Jordan canonical forms. There are $6$ where the two eigenvalues are different, and another $6$ where they're equal.

Answer 2

Let $m(x)$ be minimal polynomial then we know $m(x)$ must divide $x^3-x$.

If $ \ deg(m)=1$ then $A\in \{ 0,I,-I\}$.

And if $deg(m)=2$ then $m(x)\in \{x^2-x,x^2+x,x^2-1\}$ And hence $m$ has distinct eigenvalues which implies $A$ is diagonalizabile.