classification up to similarity of complex n-by-n matrices Classify up to similarity all 3 x 3 complex matrices $A$ such that $A^n$ = $I$.

It is an Hoffman Kunze exercise problem. It will be for $3\times 3$ matrices $A$, $A^3=I$. 
My answer is, the minimal polynomial of $A$ will divide $X^3-1=0$. Now $x^3-1=(x-1)(x-\omega)(x-\omega^2)$ where $\omega^3=1$. So the minimal polynomial can be of the forms 


*

*$m=x-a$

*$m=(x-a)(x-b)$

*$m=(x-a)(x-b)(x-c)$


Now if $m=(x-a)$,then characteristic polynomial of $A$ will be equal to $(x-a)^3$. Hence $A$ is similar to a diagonal matrix with all entries equal.
If $m=(x-a)(x-b)$, then characteristic polynomial of $A$ will be equal to  either $(x-a)^2(x-b)$ or $(x-a)(x-b)^2$. Hence $A$ is similar to a diagonal matrix with two entries equal.
If $m=(x-a)(x-b)(x-c)$, then characteristic polynomial of $A$ will be equal to  either $(x-a)(x-b)(x-c)$. Hence $A$ is similar to a diagonal matrix with all entries are unequal.
