I am working on the following problem about the orientability of real projective space.
Let $M = \mathbb{R}P^n$ with its standard atlas $(U_i,\phi_i)$ for $0 \leq i \leq n$.Calculate the determinant of the Jacobian matrix of the transition functions $ \phi_i \circ \phi^{-1}_i $ for all $0 \leq i \leq n$. Use this result to prove that $\mathbb{R}P^n$ for $n ≥ 1$ is orientable for $n$ odd, but non-orientable for $n$ even.
I calculated the Jacobian matrix which was $ \frac {-1}{x^{n+1}_i}$. How can I use this result to prove the orientability for the cases of odd and even?