Section 3.7 Solving Systems of Linear Equations by Using Matrices 293 Concepts 1. Introduction to Matrices 2. Solving Systems of Linear Equations by Using the Gauss-Jordan Method 1. Introduction to Matrices In Sections 3.2, 3.3, and 3.6, we solved systems of linear equations by using the substitution method and the addition method. We now present a third method called the Gauss-Jordan method that uses matrices to solve a linear system. A matrix is a rectangular array of numbers (the plural of matrix is matrices). The rows of a matrix are read horizontally, and the columns of a matrix are read vertically. Every number or entry within a matrix is called an element of the matrix. The order of a matrix is determined by the number of rows and number of columns. A matrix with m rows and n columns is an (read as “m by n”) matrix. Notice that with the order of a matrix, the number of rows is given first, followed by the number of columns. Determining the Order of a Matrix Determine the order of each matrix. 1 0 0 0 1 0 0 0 1 ≥ § a. b. c. d. Solution: a. This matrix has two rows and three columns.Therefore, it is a 2 3 matrix. b. This matrix has four rows and one column.Therefore, it is a 4 1 matrix. A matrix with one column is called a column matrix. c. This matrix has three rows and three columns.Therefore, it is a 3 3 matrix. A matrix with the same number of rows and columns is called a square matrix. d. This matrix has one row and three columns.Therefore, it is a 1 3 matrix. A matrix with one row is called a row matrix. Skill Practice Determine the order of the matrix. 1. 2. 3. 4. 5 10 15 A matrix can be used to represent a system of linear equations written in standard form. To do so, we extract the coefficients of the variable terms and the constants within the equation. For example, consider the system 2x y 5 x 2y 5 The matrix A is called the coefficient matrix. A c 2 1 1 2 d c 2 0.5 1 6 £ § d 34 84 £ 5 2 1 3 8 9 § £ 3a b c4 1.9 0 7.2 6.1 c 2 4 1 d ¥ 5 p 27 Example 1 m n Answers 1. 3 2 2. 1 2 3. 3 1 4. 2 2 Solving Systems of Linear Equations by Using Matrices Section 3.7
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