In this article, a new geometric r-Frank matrix is introduced, which represents an extension for the generalized Frank matrix and the Frank matrix. Subsequently, our focus is directed towards the discussion of the algebraic structure of the geometric r-Frank matrix and its reciprocal matrix and distinct representations. These efforts enable us to elucidate the abundant mathematical properties, including factorizations, determinants, inverses, and certain norms. Additionally, comparative analysis is conducted to demonstrate the bounds of the spectral norms of the geometric r-Frank matrix. Finally, as an intriguing application, to validate our norm results, the aforementioned properties of the geometric r -Frank matrix with entries being Fibonacci numbers are presented.