Matrices, Determinants, and Their Applications in Solving Systems of Linear Equations

Matrices and determinants are fundamental tools of linear algebra with extensive applications in engineering, physics, computer graphics, economics, and data science. This paper presents a comprehensive study of matrix algebra, determinant evaluation, and methods for solving systems of linear equations, suitable for polytechnic and diploma-level students. The paper covers matrix types and operations (addition, multiplication, transpose, inverse), determinant evaluation using cofactor expansion and row reduction, three methods for solving systems of linear equations (Cramer's Rule, Gaussian Elimination, and the Matrix Inversion method), and an introduction to eigenvalues and eigenvectors with engineering applications. Each method is illustrated with fully worked numerical examples and results are presented in tabular form for easy comparison. Applications to structural analysis (simultaneous load equations), electrical circuit analysis (mesh current method), and network flow problems are demonstrated. Verification of solutions by back-substitution is carried out for all worked examples. The paper demonstrates that matrix methods provide a systematic, organised approach to solving linear systems that are far superior to conventional algebraic elimination for systems of three or more variables.