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

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 Aug 13, 2021 at 18:58
• Does this answer your question? $A$ is diagonalizable and $A^3 = A^2$ – found with Approach0 Aug 13, 2021 at 19:08
• Another approach is possible if you know the minimal polynomial for a diagonalizable matrix has no repeated roots… Aug 13, 2021 at 19:09

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