Prime number analysis in matrices – A case study on linear dependence

Prime numbers constitute the structural foundation of number theory and continue to be acentral focus of study and research efforts today. Significant open problems remain in the field,serving as major drivers for the advancement of mathematical tools and the renewal of intellectualinspiration dedicated to this area. This work adopts an exploratory approach, providing an entry pointinto the principal connections between matrices and applications of prime numbers. It aims toestablish a theoretical basis grounded in known advances regarding the interplay between matrixalgebra and number theory, as well as to present a specific analysis of two particular cases identified innumerical investigations of the topic. Finally, the study concludes with an examination of the resultsand potential directions for future research.