Applied Engineering Analysis

Chapter 4Linear Algebra and Matrices

Chapter Learning Objectives

• Linear algebra and its applications
• Forms of linear functions and linear equations
• Expression of simultaneous linear equations in matrix forms
• Distinction between matrices and determinates
• Different forms of matrices for different applications
• Transposition of matrices
• Addition, subtraction and multiplication of matrices
• Inversion of matrices
• Solution of simultaneous equations using matrix inversion method
• Solution of large numbers of simultaneous equations using Gaussian elimination method
• Eigenvalues and eigenfunctions in engineering analysis

4.1 Introduction to Linear Algebra and Matrices

Linear algebra is concerned mainly with systems of linear equations, matrices, vector space, linear transformations, eigenvalues, and eigenvectors. It is widely applied to other areas of mathematics and to other various disciplines of engineering and science (Hogben, 1987).

Linear functions are often used to represent straight lines. For example, the function is used to represent a straight line in the x–y plane, in which m and b are constants, with m representing the “slope” of the straight line and b the intercept of the line with the y-axis as illustrated in Figure 4.1. Equations involving linear functions are linear equations.

Figure 4.1 Linear functions.

Engineers need to deal with various types of equations; ...