For any odd prime number \(p\), the Maillet determinant \(D(p)\) is related to the relative class number of the cyclotomic field \(\mathbb{Q}(\zeta_p)\) by the formula of Carlitz and Olson: \[ D(p)= (-p)^{(p-3)/2} h^- (\mathbb{Q}(\zeta_p)). \] This has been generalized to imaginary abelian fields by Tateyama, Girstmair, etc. \dots The determinant of the Demjanenko matrix \(M(p)\) is analogously related to \(h^-(\mathbb{Q} (\zeta_p))\). This has been generalized to imaginary abelian fields by Tsumura, Hirabayashi, etc. \dots{} In this note, the author introduces the cyclotomic \(\mathbb{Z}_p\)-extension \(K_\infty= \bigcup_m K_m\) of an imaginary abelian field with conductor \(dp\), \(p\nmid d\) and generalized Demjanenko matrices \(\Delta(K,\ell,m)\), \((\ell,dp)=1\); the main result is a formula relating \(\det \Delta(K,\ell,m)\) to \(h^-(K_m)\).