Show that $A=cI$ for c non-zero real number.

Question

We have A 2×2 matrix with real numbers. We know that there is an odd n such that $A^{n}=I$. Show that $A=cI$ for c non-zero real number.
We can find easily that $det(A)=1$. I took the polynomial $P(x)=x^{n}-1$. So $P(x)=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$
A is solution for P(x). So the minimal polynomial $m_{a}|P(x)$. We find that $m_{a}$ can be $(x-1)$ wich shows $A=I$. From here I dont know how to continue with that cyclotronic polynomial.