If A is a square matrix of order 3 and I is Identity matrix of order 3 such that $A^3-2A^2-A+2I=0,$  then A could be

(a,b,d) it is clear that  $A=IandA=2I$ satisfy the given equation $A^3-2A^2-A+2I=0$ and the characteristic equation of the matrix in (c) is

$\left[\begin{array}{ccc}2-\lambda & -1& 2\\ 1& -\lambda & 0\\ 0& 1& -\lambda \end{array}\right]=0$

$\Rightarrow {\lambda }^3-2{\lambda }^2+\lambda -2=0$

Given  $A^3-2A^2+A-2I=0$  

$\ne A^3-2A^2-A+2I=0$ and the characteristic equation of the matrix in (d) is

$$\begin{gathered} \Rightarrow \left[\begin{array}{ccc}2-\lambda & 1& -2\\ 1& -\lambda & 0\\ 0& 1& -\lambda \end{array}\right]=0 \\ {\lambda }^3-2{\lambda }^2-\lambda +2=0 \\ giving A^3-2A^2-A+2I=0 \end{gathered}$$