Basic Linear Algebra In this chapter,we study the topics in linear algebra that will be needed in the rest of the book. We begin by discussing the building blocks of linear algebra:matrices and vectors.Then we use our knowledge of matrices and vectors to develop a systematic procedure(the Gauss- Jordan method)for solving linear equations,which we then use to invert matrices.We close the chapter with an introduction to determinants. The material covered in this chapter will be used in our study of linear and nonlinear programming. 2.1 Matrices and Vectors Matrices DEFINITION■ A matrix is any rectangular array of numbers. For example, [ [21] are all matrices. If a matrix A has m rows and n columns,we call A an m x n matrix.We refer to m X n as the order of the matrix.A typical m X n matrix A may be written as a11a12… a21 a22 A DEFINITION■ The number in the ith row and ith column of A is called the ijth element of 4 6 and is written ay For example,if [12 31 A=45 6 789 then a11=1,a23=6,anda31=7.Basic Linear Algebra In this chapter, we study the topics in linear algebra that will be needed in the rest of the book. We begin by discussing the building blocks of linear algebra: matrices and vectors. Then we use our knowledge of matrices and vectors to develop a systematic procedure (the Gauss– Jordan method) for solving linear equations, which we then use to invert matrices. We close the chapter with an introduction to determinants. The material covered in this chapter will be used in our study of linear and nonlinear programming. 2.1 Matrices and Vectors Matrices DEFINITION ■ A matrix is any rectangular array of numbers. ■ For example, , , , [2 1] are all matrices. If a matrix A has m rows and n columns, we call A an m n matrix. We refer to m n as the order of the matrix. A typical m n matrix A may be written as A DEFINITION ■ The number in the ith row and jth column of A is called the ijth element of A and is written aij. ■ For example, if A then a11 1, a23 6, and a31 7. 3 6 9 2 5 8 1 4 7 a1n a2n    amn       a12 a22    am2 a11 a21    am1 1 2 3 6 2 5 1 4 2 4 1 3