5.3 Matrices and Determinants Objectives:After completing this section, you should be able to solve systems of linear equations using the following methods:1.Matrices and Gaussian elimination 2.Matrices and Gauss-Jordan elimination 3.Cramer’s ruleIn Sections 1 and 2 we learned how to solve systems of linear equations in two and three variables using elimination by addition and its variant, Gaussian elimination. We can organize more efficiently our work for solving linear systems using these methods, and two other methods you will see later, by introducing a mathematical form called a matrix. Corresponding to a linear system of equations is an augmented matrixwhich contains all of the information in the system without the x, y, and variable labels to carry around. In this section we will revisit Gaussian elimination but this time using augmented matrices. Keep This in Mind A matrixis a rectangular array of numbers written within brackets that contains rows and columns. If m and nare positive integers, an m xn(read “m byn”) matrix is a rectangular array of numbers mnmmnnaaaaaaaaa212222111211with mrows and n columns in which each number ijais referred to as an entrythe matrix in the ithrow and jth column. The expression m xnis called the sizedimension of the matrix. A matrix with nrows and ncolumns is called a square matrix of order n. A matrix with only one is called a column matrix, and a matrix with only one row is called a row matrixRow 1 Row 2 Row Col. 1 Col. 2 Col. zof or . mn
Unit 5. Systems of Equations and Inequalities Section 3 page 2 _____________________________________________________________________________________ Example 5.3.1 Below are two matrices:702431A2503861015104BWe usually name a matrix by a single capital letter, such as A, B, C, and so on. Matrix A above has six entries arranged in two rows and three columns, while matrix Bhas 12 entries arranged in three rows and four columns. So we call A as a 2 x 3 matrix and B as a 3 x 4 matrix. In matrix Awe have 723aand 312a, while in matrix Bwe have 623band .331bMatrices Associated with Systems of Linear Equations Consider the following system of linear equations in the workstation problem of Example 5.1.2in Section1: .163832yxyxRelated to this system are the following matrices: 1132163816113832The augmented matrixis what we need for Gaussian elimination. It is called “augmented” because it includes the right-hand side constants of the equations. The other matrices will be used later in this section. The augmented matrix contains the essential parts of the linear system –both the coefficients and the constants. The vertical bar is included only as a visual aid to help us separate the coefficients from the constants.