A New Year, A New Hard Problem! 5 of 5 (The Hardest One)

For an integer $n\ge 3$ , let $\theta = \dfrac {2\pi}n$ . Evaluate the determinant of the $n\times n$ matrix $I+A$ , where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk} = \cos (j\theta+k\theta)$ for all $j,k$ .