GCD Determinant Tango

Consider the matrix $X \in \mathcal{S}$ where $X_{ij} \; = \; \left\{\begin{array}{lll} 1 & \hspace{1cm} & i \le j \\ -1 & & i > j \end{array}\right.$ If we add the first row of $X$ to the second, third, fourth, ..., seventeenth rows of $X$ , we deduce that $|X| = |Y|$ , where $Y$ is the triangular matrix $Y_{ij} \; = \; \left\{ \begin{array}{lll} 1 & \hspace{1cm} & i = 1 \\ 2 & & 2 \le i \le j \\ 0 & & i \ge 2\,,\, j < i \end{array}\right.$ and hence $|X| = |Y| = 2^{16}$ . On the other hand, for any $A \in \mathcal{S}$ , adding or subtracting the first row of $A$ from the other rows of $A$ gives us that $|A| = |B|$ , where $B \; = \; \left(\begin{array}{cccc}\pm1 & \pm1 & \cdots & \pm1 \\ 0 & & & & \\ \vdots & & C & \\ 0 & & & \end{array}\right)$ where each entry of $C$ is either $2$ , $0$ or $-2$ . Thus each row of $B$ , except for the first one, is divisible by $2$ , and hence $|A|$ is divisible by $2^{16}$ .

Thus we deduce that the greatest common divisor all the determinants of matrices in $\mathcal{S}$ is $2^{16} = \boxed{65536}$ .