Jordan form of matrix using the minimal polynomial

Question

I've been studying Linear Algebra for the past few days and was busy with the topic of jordan matrices. I was making some exercices and came across the following one: Let $X^4 - 2X^3 + X^2$ be the minimal polynomial of some matrix A in $\mathbb{C}^{4x4}$, determine the jordan form of $A$ and $A^3$. I could determine the Jordan form of $A$ and got the following result $$\begin{pmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 1 & 1\\ \end{pmatrix}$$ Well I think this is it, I'm not sure tho. Following this I though the jordan form of $A^3$ would just be the one from $A$ but multiplied 3 times, however this didn't result in a jordan form at all and thus I got stuck. I thought that i could maybe determine the minimal polynomial of the matrix $A^3$ knowing the minimal polynomial of that from $A$, but haven't been able to progress with this problem. Does anyone have a tip? Any help would be greatly appreciated :))

Answer 1

The minimal polynomial of a matrix $A$ and $A^3$ need not be same in general. For example, consider $A=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{pmatrix}.$ Then the minimal polynomial of $A$ is $x^3.$ However, $A^3=0$ matrix and the minimal polynomial of $A^3$ is $x.$