Show that the trace of A is less than n

Question

Let $A$ be an $n\times n$ matrix with complex entries such that $A^k=I_n$ for some positive integer $k$. Show that the trace of $A$ satisfies $$|tr(A)| \leq n.$$

I have no idea how to approach this question. Please help.

Answer 1

$A^k=I$ means its minimal polynomial divides $x^k-1$ and thus has distinct roots. The trace is the sum of those roots (the sum of the eigenvalues). As roots of $x^k-1$ those are all roots of unity so lie on the unit circle in $\Bbb C$. So you need to show that the sum of $n$ roots of unity is no bigger than $n$. That's a sum of $n$ things on the unit circle in $\Bbb C$ so it's geometrically obvious that it cannot have absolute value greater than $n$.