How to prove that if $T$ is diagonalizable and $T^3=T^2$, $T^2=T$?

Question

For my linear algebra homework, I need to prove that if $T$ is a diagonalizable linear map and $T^3=T^2$, then $T^2=T$. I tried proving that $T$ is invertible (so that I can multiply both sides of $T^3=T^2$ by $T^{-1}$, but apparently diagonalizability does not imply invertibility (see Can a matrix be invertible but not diagonalizable? in the comment section of the first answer).

Could you help me?

This is quite simple - all you have to do is write $T = SDS^{-1}$ from definition where $D$ is a diagonal matrix. The rest is easy — Jakobian, Aug 13 '21 at 18:58
Does this answer your question? $A$ is diagonalizable and $A^3 = A^2$ – found with Approach0 — Martin R, Aug 13 '21 at 19:08
Another approach is possible if you know the minimal polynomial for a diagonalizable matrix has no repeated roots… — Thomas Andrews, Aug 13 '21 at 19:09

score 2 · Accepted Answer · answered Aug 13 '21 at 19:00

If $T$ is diagonalizable, then you have a diagonal matrix $D$ and an invertible matrix $P$ such that $T=PDP^{-1}$.
Now you have $$T^3=T^2$$ $$PDP^{-1}PDP^{-1}PDP^{-1}=PDP^{-1}PDP^{-1}$$

$$PD^3P^{-1}=PD^2P^{-1}%$$ $$D^3=D^2$$

Now, you can't just apply $D^{-1}$ as that might not exist, but what is the formula entrancewise for $D^n$? from that you should be able to conclude what $D$ has to look like and finish

How to prove that if $T$ is diagonalizable and $T^3=T^2$, $T^2=T$?

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