Identity is the only real matrix which is orthogonal, symmetric and positive definite 
Show that identity is the only real matrix which is orthogonal, symmetric and positive definite

All I could get using above information was that $A^2=I$, hence it is its own inverse.
Using the fact that $A$ is positive-definite, I got that all diagonal entries will be greater than $0$, but how does that help?
Edit:
As $A$ satisfies $(x^2-1)=0$, therefore the minimal polynomial will divide this. Therefore, the minimal polynomial will have $(x-1)$ or $(x+1)$ or both as a factor, as $A$ is positive definite, so $-1$ can't be an eigenvalue, therefore we get that $1$ is an eigenvalue of $A.$
I'm not sure, if this helps, though.

An alternative approach: as $A^2=A^TA=I$, the minimal polynomial of $A$ is either $x-1,\ x+1$ or $x^2-1$. Yet all eigenvalues of $A$ are positive (because $A$ is positive definite). So, the minimal polynomial must be ...
