If $\mathbf A$ is an $n \times n$ matrix such that $\mathbf A^6 = \mathbf I_n$ (the identity matrix), is it true that either $\mathbf A^3 = \mathbf I_n$ or $\mathbf A^3 = \mathbf -I_n$?
I'm struggling to solve this question.
I re-wrote $\mathbf A^3$ as $\mathbf B$, which gives $\mathbf B^2 = \mathbf I_n$. I do know that, if $\mathbf B^2 = \mathbf I_n$, then it is not necessarily true that $\mathbf B = \pm\mathbf I$ (I had a look at the following link, though I haven't fully understood the explanations given: If $A^2 = I$ (Identity Matrix) then $A = \pm I$).
Can I (and if so, how do I) apply this fact to the case where $\mathbf B$ is the cube of another matrix $\mathbf A$?