Abstract: We investigate the family of structured matrices $$ T_c^{(n)} $$ defined by a geometric sum $$ S_n=\sum_{k=0}^{n}c^k $$ in the first entry and a descending power pattern in subsequent rows. A closed formula for $$ \det\left(T_c^{(n)}\right) $$ is derived, showing that for integer $$ c $$ the determinant is nonzero exactly when $$ c\notin\{1,-1\} $$ and equals 1 only for $$ c=0 $$ (if $$ n\geq 3 $$ ). The matrices exhibit a reciprocal symmetry expressed through a scaled Hyper Gpair relation: there exist diagonal matrices $$ D_1,\;D_2 $$ and a third diagonal matrix $$ \Gamma(c) $$ such that $$ \left(T_c^{(n)}\right)^{-T} = \Gamma(c)D_1T_{1/c}^{(n)}D_2. $$ This generalizes the classical notion of Hyper Gmatrices, where $$ \Gamma(c) $$ would be a scalar, and reveals a richer algebraic structure linking $$ T_c^{(n)} $$ with its reciprocal parameter counterpart. The results unify combinatorial determinant evaluation, modularlike matrix symmetries, and the theory of scaled Gpairs.

Keywords: Structured matrices, Determinant formulas, Hyper G-matrices, Scaled reciprocal relations, Matrix symmetries, Integer parameter matrices, Block matrices, Diagonal congruences

Pages: 240-251