TITLE: $A$ is diagonalizable and $A^3 = A^2$
QUESTION [6 upvotes]: If $A$ is diagonalizable and $A^3 = A^2$. Is it necessary true that $A^2 = A$?

REPLY [7 votes]: The polynomial $P=X^3-X^2=X^2(X-1)$ satisfies $P(A)=0$.
Now, if $A$ is diagonalizable, the roots of the minimal polynomial $\mu_A$ of $A$ are of multiplicity one. Since $P(A)=0$, then $P$ is a multiple of the minimal polynomial of $A$, hence $\mu_A$ must divide $X(X-1)$ hence we must have $A(A-I)=0$, i.e., $A^2=A$.