I tried using the definition of affine transformation, where I got the following:
$f^2(X) = f(f(X)) = C + Mf(X) = C + M(C + MX) = (I + M)C + M^2X = X \Rightarrow \left\{ \begin{aligned} (I+M)C = 0 \\ M^2=1 \end{aligned} \right. $
There, I get that either $C=0$ or $M=-I$.
If $C=0$, then $f(X) = MX$, so $M = I$, and if $M = -I$, $f(X) = C - X$, so $C=2X$. But I'm not so sure that what I wrote is a good proof (or correct, in that regard). How do I do it, then?
It doesn't need to be an unique fixed point. The result says that there is "at least" one fixed point.
P.S.: How do I do it if instead of having $f^2$ I have $f^n$, for $n\in\mathbb N$?