Determinant Formulas for Weighted Greatest Common Divisor Matrices

In a companion paper [10], we established the Fundamental Isomorphism Theorem for Weighted Greatest Common Divisor (WGCD) matrices: every WGCD matrix is precisely the classical GCD matrix constructed from its diagonal entries. In this paper, we leverage this isomorphism to derive explicit determinant formulas for WGCD matrices. We prove that if $\Sigmamat$ is a WGCD matrix with diagonal entries $s_0, \ldots, s_n$, then the determinant satisfies a weighted generalization of Smith's determinant formula. Specifically, for factor-closed diagonal sets, $\det(\Sigmamat) = \prod_{i=0}^n \phi(s_i)$; for general sets, we provide a complete summation formula via the Cauchy-Binet theorem. We establish Li's lower bound $\det(\Sigmamat) \ge \prod_{i=0}^n \phi(s_i)$, with equality if and only if the diagonal set is factor-closed. We also derive Hadamard-type upper bounds and characterize the determinant in terms of the generalized Euler's totient function for gcd-closed sets. Our results generalize the classical theorems of Smith (1876), Beslin and Ligh (1989), and Li (1990) to the weighted setting. Numerical examples and algorithmic implications are discussed.