Algebraic Structure of Nilpotent and Idempotent Matrices

Group theory has significantly advanced the study of algebraic structures by examining examples that satisfy group axioms and by introducing modifications to define new types of groups. This research focuses on the algebraic structures formed by sets of nilpotent matrices and idempotent matrices. The first part investigates the set of nilpotent matrices under matrix multiplication, denoted as , analyzing its compliance with group properties. The second part explores the algebraic structure of idempotent matrices with multiplication as the operation, represented as Through rigorous examination of closure, associativity, identity elements, and inverses within these sets, this study reveals that while forms a semigroup due to lack of identity and inverses, constitutes a monoid but not necessarily a group because some elements lack inverses. Additionally, intersections and unions between these sets are discussed to highlight their structural properties. These findings contribute to a deeper understanding of matrix-based algebraic systems and provide groundwork for further exploration in abstract algebra.