Can this determinant ever vanish? Let $a_j=1 +2\cos\left(\frac{2\pi j}q\right)$ for $j=1,\dots,q$. Then consider the Hermitian matrix:
$$A_q = \begin{pmatrix} 
           a_1 &      1 &      0 &      0 & \ldots &       0 & 1 \\ 
             1 &    a_2 &      1 &      0 & \ldots &       0 & 0 \\ 
             0 &      1 &    a_3 &      1 & \ldots &       0 & 0 \\
             0 &      0 &      1 &    a_4 & \ldots &       0 & 0 \\ 
        \vdots & \vdots & \vdots & \vdots & \ddots &  \vdots & \vdots \\
             0 &      0 &      0 &      0 & \ldots & a_{q-1} & 1 \\ 
             1 &      0 &      0 &      0 & \ldots &       1 & a_q
        \end{pmatrix}$$
Using Mathematica, I find:
$$\begin{align}
\det(A_2)&=-4\\
\det(A_3)&=-1\\
\det(A_4)&=-7\\
\det(A_5)&=-\frac52(-5+\sqrt5)
\end{align}$$
Is there any general formula that describes this determinant? And, more importantly, can the determinant ever be zero? The one answer I got so far, suggests no, but it is of course not a proof.
This problem is motivated by a quantum mechanics problem, where the Hermitian matrix describes an observable in QM. In particular, it originates from the study of an electron on a lattice in a commensurable magnetic field.

Because of the $1$'s in the upper-right and lower-left corners, I didn't have any luck trying to derive a recurrence relation. But using Matlab, I was able to at least get the first several terms:




$q$
$\text{det}\left(A_q\right)$




$1$
$3$


$2$
$-4$


$3$
$-1$


$4$
$-7$


$5$
$\frac{5-\sqrt{5}}{4}$


$6$
$5$


$7$
$12.95944337$


$8$
$-5.05887450$


$9$
$-12.24073273$


$10$
$-26.54915028$




The latter four decimal expansions don't even appear in OEIS, so as saulspatz notes in the his comment, "pretty" closed-form expressions are unlikely.
In terms of asympotics, plotting the first 1000 terms shows an approximate exponential relationship:
$\begin{align}\left|\text{det}\left(A_q\right)\right|\sim \exp\left(\beta_0+\beta_1 q\right) \hspace{1 em} \text{where} \hspace{1 em} &\beta_0 = -0.0103 \pm 0.09\\
&\beta_1 = 0.2514 \pm 0.0004\end{align}$
$\text{sgn}\left(\text{det}\left(A_q\right)\right)$ also appears to be somewhat periodic; the sign changes every $3$ or $4$ iterations from $2\leq q\leq 1000$.
This is far from a complete answer, but I hope it's enough to serve as a branching off point.
