Abstract This paper presents a comprehensive exploration of matrix theory, focusing on the classification of matrices, determinant computation, and fundamental properties. Matrices serve as essential mathematical structures in linear algebra with widespread applications across physics, engineering, computer science, and economics. This study systematically examines various matrix types including square, rectangular, diagonal, symmetric, and special matrices, followed by an in-depth analysis of determinants and their algebraic properties. The paper establishes theoretical foundations while demonstrating practical computational methods, providing a rigorous treatment suitable for advanced undergraduate and graduate-level study in mathematics and related disciplines.