The objective of this chapter is to prepare for the use of MATLAB to find solutions for a system of linear equations. The important concepts include the inverse matrix, solution of a system of linear equations, least squares method, and MATLAB’s “left-division,” which is essentially an implementation of the least squares method. MATLAB functions are very useful and efficient in the job. For real problems, the equations are often highly overdetermined, which occurs when there are many more equations (or measurements) than the number of unknowns, and thus it requires the use of the least squares method to solve the solution in a statistical sense. The contents in this chapter lay out a foundation for several later chapters because many theories and methods are related in one way or the other to the solution of a system of linear equations, e.g. Fourier analysis and harmonic analysis.

This chapter provides a basic introduction to matrices, including the following: scaling, transposing, adding, and subtracting matrices; multiplying matrices and applications; finding determinants and inverses of 2 x 2 matrices; and solving systems of equations by matrix inversion. Applications of matrix algebra, including applications to cryptography and to Leontief economic models, are discussed.A method for finding inverses of larger matrices using elimination is developed in a series of exercises.