Solve equation $X^r=Id$ in $\operatorname{SL}_n(\Bbb F_q)$

Question

Let $q$ be an odd prime power. Solve equation $X^r=Id$ in $\operatorname{SL}_n(\Bbb F_q)$ where $r$ divides

$$|\operatorname{SL}_n (\Bbb F_q)| = \frac{1}{q-1} \prod_{i=0}^{n-1}(q^n-q^i)$$

A solution $X$ should be diagonalizable. Hence the eigenvalues of $X=PDP^{-1}$ should be all $r$-th roots of unity lying on the diagonal matrix D. I don't know how to proceed from here, How can we fully describe $P$ and therefore describe a solution $X$.

Edit

My statement above about the fact that solutions should be diagonalisable is false. I had also in mind a related question

• Is it possible to link the number of solutions to the number $r$?

Many thanks for your help.