If $A^3=I_n$ (the identity matrix), then what are the possible minimal polynomials for $A$?

Question

I just got done with an exam and one question was to determine the possible minimal polynomials of $A$, if $A^3$ is the identity matrix. Note that $A$ is just some square matrix over $\mathbb{Q}$. My thoughts were that since $$A^3-I_n=0$$ then $A$ is a root of $x^3-1$. But this factors as $(x-1)(x^2+x+1)$, so $A$ must be a root of one of those factors. Thus the possible options for the minimal polynomial of $A$ over $\mathbb{Q}$ are $x-1$ and $x^2+x+1$. Am I on the right track?

Answer 1

You're right that $A$ fulfills $x^3-1$, and the minimal polynomial must therefore be a factor of $x^3-1$. You are missing two things, however:

1. The minimal polynomial could also be $x^3-1$
2. You need to demonstrate that all of these are actually possible. For instance by presenting, for each polynomial, a matrix with that polynomial as minimal polynomial

Answer 2

I think you're right, since the mononic polynomial $x^2+x+1$ doesn't splits over $\mathbb{Q}$ and $x-1$ splits over that field. Something that would be more efficient is having the characteristic polynomial so by Calley-Hamillton Theorem, you would know at least factors of your minimal polynomial but anyway those product of irreducible (monic) polynomials divides the possible minimal and of course the characteristic.