TITLE: classification up to similarity of complex n-by-n matrices
QUESTION [5 upvotes]: Classify up to similarity all 3 x 3 complex matrices $A$ such that $A^n$ = $I$.

REPLY [3 votes]: In fact, one does not need to know the characteristic polynomial in this case. Let the minimal polynomial be $p$, then $p\mid (x^3-1)$. It is important to see that $x^3-1$ has three distinct roots in $\mathbb{C}$. Hence $p$ cannot have repeated roots in $\mathbb{C}$. Thus $A$ must be diagonalizable over $\mathbb{C}$, with each diagonal entry a root of $x^3-1$. Hence $A$ is similar to 
\begin{equation}
\begin{pmatrix}
a & 0 & 0 \\
0 & b & 0 \\
0 & 0 & c
\end{pmatrix},
\qquad
a,b,c \in \{1,e^{i2\pi/3},e^{-i2\pi/3}\}
\end{equation}