Determinants: Theory, Properties, and Applications

Abstract Determinants are a fundamental concept in linear algebra that provide crucial insights into the properties and behavior of square matrices. Originating from the need to solve systems of linear equations, determinants have evolved into a powerful mathematical tool with wide-ranging theoretical and practical applications. This paper examines the concept of determinants, including their definition, computational techniques, and key properties. It highlights the role of determinants in determining matrix invertibility, solving linear systems through Cramer’s Rule, and calculating eigenvalues. Furthermore, the study explores the geometric interpretation of determinants as measures of area and volume, offering a meaningful connection between algebra and geometry. Applications across various disciplines such as physics, engineering, computer science, and economics are also discussed to demonstrate their interdisciplinary significance. Despite computational challenges for large matrices, determinants remain an essential component of modern mathematical analysis and problem-solving. The paper concludes by emphasizing the continued relevance of determinants in both theoretical research and applied sciences.