This study introduces a new mathematical structure called a “single-valued trivalent logic-based fuzzy rhotrix” (STFR). An STFR combines a r x r matrix and a (r -1) x (r -1) matrix with a membership function. This membership function takes its values from a set based on trivalent logic (a logic system with three possible truth values). The paper explores the fundamental characteristics and properties of these newly defined STFRs, presenting the findings. Furthermore, the research initiates an investigation into the concepts of determinant functions, codeterminant functions, and an associated “index” within the context of STFRs.