A question says, write down the possible minimal polynomials which have characteristic polynomial $(1-x)(1-x^3)$, and for each possibility find a specific example of a matrix having this minimal polynomial.
Previously we're told that a matrix with minimal polynomial $x^2 + x + 1$ is $\begin{pmatrix}0&-1\\1&-1\end{pmatrix}$.
So factorising $(1-x)(1-x^3)$, gives, $-(x-1)^2(x^2 + x + 1)$.
I understand that a two possible minimal polynomial are $(x-1)^2(x^2 + x + 1)$ and $(x-1)(x^2 + x + 1)$ with matrices $\begin{pmatrix}1&1&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&-1\end{pmatrix}$ and $\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&-1\end{pmatrix}$ respectively, but why can't we have a minimal polynomial $(x^2 + x + 1)$ and find a specific matrix? (I have the answers to this question by the way and it doesn't give that as a minimal polynomial).