how can we find the number of matrices with real entries of size $9 \times 9$ (up to similarity) such that $A^{2}=I$?
I first thought about the following:
Notice $A$ satisfies the polynomial $f(t)=t^{2}-1$ hence its minimal polynomial divides $(t-1)(t+1)$.
So its characteristic polynomial is of the form $p(t)=(t-1)^r(t+1)^j$ where $r+j = 9$, right? Then I'm not sure what to do, I tried to consider the rational canonical form but in order to do this we need to know the minimal polynomial right? because in the rational canonical form the last term in the array is exactly the minimal polynomial, how to find it?
Can you please help?