Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for $k=0,\dots,l-1$. Define $M(\alpha,\beta)$ to be the $l\times l$ matrix whose entry in $i$th row and $j$th column is $\alpha_{i-1}^{j-1}\beta_{j-1}^{i-1}\zeta^{(i-1)(j-1)}$.

**Question:** Is $M(\alpha,\beta)$ nonsingular for all $\alpha,\beta$?

If $\alpha$ or $\beta$ is constant, then since $$ \alpha_i^j\beta_j^i\zeta^{ij}=\alpha_i^j(\beta_j\zeta^j)^i=\beta_j^i(\alpha_i\zeta^i)^j, $$ $M(\alpha,\beta)$ is equivalent to a Vondermonde matrix and thus nonsingular.

allpossibilities for $\alpha$ and $\beta$, the answer is affirmative for all $\ell\le12$. $\endgroup$4more comments