Determinant Symmetry and Scaled Hyper G-Pair Relations of Tc(n) Matrices
WSEAS TRANSACTIONS ON CIRCUITS AND SYSTEMS, cilt.25, ss.240-251, 2026 (Scopus)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 25
- Basım Tarihi: 2026
- Doi Numarası: 10.37394/23201.2026.25.21
- Dergi Adı: WSEAS TRANSACTIONS ON CIRCUITS AND SYSTEMS
- Derginin Tarandığı İndeksler: Scopus, INSPEC
- Sayfa Sayıları: ss.240-251
- Karadeniz Teknik Üniversitesi Adresli: Evet
Özet
We investigate the family of structured matrices T (n) c defined by a geometric sum Sn = ∑n k=0 ck in the first entry and a descending power pattern in subsequent rows. A closed formula for det(T (n) c ) is derived, showing that for integer c the determinant is nonzero exactly when c /∈ {1, −1} and equals 1 only for c = 0 (if n ≥ 3). The matrices exhibit a reciprocal symmetry expressed through a scaled Hyper Gpair relation: there exist diagonal matrices D1, D2 and a third diagonal matrix Γ(c) such that
(T (n)
c )−T = Γ(c) D1 T (n)
1/c D2.
generalizes the classical notion of Hyper Gmatrices, where Γ(c) would be a scalar, and reveals a richer algebraic structure linking T (n) c with its reciprocal parameter counterpart. The results unify combinatorial determinant evaluation, modularlike matrix symmetries, and the theory of scaled Gpairs.