In my notes, I have a $4\times 4$ complex matrix $A$ with the following properties. The characteristic polynomial of $A$ is $(x-1)(x+1)^3$, and the geometric multiplicity of $-1$ is $2$. That is all that is known about $A$.
In determining the minimal polynomial, the possibility $(x-1)(x+1)$ is eliminated because "the invariant factors must multiply to the characteristic polynomial, but there are only two invariant factors."
What is telling us that there are only two invariant factors?
I'm not really sure if this is what you mean, but the minimal polynomial is going to be $(x-1)(x+1)^2$. The factor $x-1$ has to be there, because you have an eigenvector for eigenvalue $1$. Now, for the eigenvalue $-1$ you have the generalised eigenspace of dimension $3$ (that's what the factor $(x+1)^3$ in the characteristic polynomial tells you), and there are two linearly independent eigenvectors (that's the geometric multiplicity). So, there are two Jordan boxes in the Jordan form of $A$, and their sum of dimensions is $3$. This can only happen if one of them is $1 \times 1$, and another is $2 \times 2$. It follows that you need the factor $(x+1)^2$ in the minimal polynomial (to kill the $2 \times 2$ box). Hence the final answer $(x-1)(x+1)^2$.
