SOLVING A SYSTEM OF LINEAR EQUATIONS USING CRAMER'S, GAUSS'S, AND MATRIX METHODS

Solving systems of linear equations is a fundamental task in mathematics, engineering, and applied sciences. This study explores three classical and widely used methods for solving such systems: Cramer's Rule, Gaussian Elimination, and matrix (algebraic) methods. Cramer's Rule utilizes determinants to find unique solutions of square systems, offering a straightforward approach for small-sized systems but becoming computationally impractical for larger ones due to the factorial growth in complexity. Gaussian Elimination, on the other hand, provides a more scalable algorithm by transforming the augmented matrix of the system into row echelon form, allowing for back-substitution to find solutions. The matrix method involves representing the system in the form AX = B, and if matrix A is invertible, the solution is obtained as X = A⁻¹B. This method is particularly powerful when dealing with multiple systems sharing the same coefficient matrix. The paper provides a comparative analysis of these methods based on computational efficiency, ease of implementation, and practical applicability. Examples and step-by-step solutions are included to illustrate the advantages and limitations of each approach. The study aims to help students and researchers select the most appropriate method depending on the characteristics of the linear system at hand. Keywords: Linear equations, Cramer's Rule, Gaussian elimination, matrix methods, system of equations, linear algebra, determinant, inverse matrix, row echelon form, back-substitution, matrix operations, numerical methods, computational efficiency, algebraic methods, solution techniques.