Question #41267

This question tests the student's ability to recognize hidden linear dependency in matrices constructed from binomial expansions of complex exponentials.

The primary trap lies in the Euler's form representation: students often attempt brute-force expansion or complex row operations, failing to realize that $b_{jk} = (\omega^j + \omega^{-k})^3$ expands into exactly 4 separable terms.

This structural property restricts the rank of any such matrix to at most 4, immediately rendering the $5 \times 5$ determinant $\det(B)$ zero.

Furthermore, for the $4 \times 4$ matrix $A$, the factorization into Vandermonde matrices tests the deep connection between cyclotomic polynomial discriminants and matrix determinants, punishing those who do not leverage the roots of unity algebraically.