A Cartan Matrix is a mathematical tool used in the study of Lie algebras and algebraic groups. It is a square matrix that encodes information about the relationships between simple roots of a Lie algebra. Each entry in the matrix indicates how the corresponding simple roots interact, helping to classify different types of Lie algebras. The Cartan Matrix is essential in understanding the structure of semisimple Lie algebras, which are fundamental in various areas of mathematics and theoretical physics. It plays a crucial role in the classification of these algebras, linking them to associated Dynkin diagrams and facilitating the study of their representations and symmetries.