Intriguing Determinant

Geometry Level 5

Let \(A_n=(a_{ij})\) be the \(n\times n\) matrix with \(a_{ii}=2\cos\left(\dfrac\pi{180}\right)\) , \(a_{ij}=1\) if \(|i-j|=1\) and \(a_{ij}=0\) otherwise, for all \(n\). Find the maximal value of \(\det(A_n)\), for all positive integers \(n\).

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