If a $3\times 3$ matrix satisfies $A^3-A^2-A+I=0$, then it is not necessarily diagonalisable.

If a $3\times 3$ matrix satisfies $A^3-A^2-A+I=0$, then it is not necessarily diagonalisable.

I have a counterexample $\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}$. But I was hoping for some direct proof. Any hints in this direction would be helpful. Thanks in advance.

$A^3-A^2-A+I=0$ iff \begin{align} 0&=(A^3-A)-(A^2-I) \\ &=A(A^2-I)-(A^2-I) \\ &=(A-I)^2(A+I). \end{align} The minimal polynomial for $A$ could be any one of the following: $$ x-1,(x-1)^2,x+1,(x+1)(x-1),(x+1)(x-1)^2. $$ Recall that $A$ is diagonalizable iff the minimal polynomial for $A$ has no repeated factors. So, there are examples of $A$ that are not diagonalizable, and there are examples that are diagonalizable. Examples of both are easily constructed in Jordan canonical form.