## Finite Order Matrix and its Trace

## Problem 28

Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that

**(a)** $|\tr(A)|\leq n$.

**(b)** If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$.

**(c)** $\tr(A)=n$ if and only if $A=I_n$.