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Com/pic/ larsonELA6e. ΄ 2 1 0 Ϫ1 0 1 4 3 Ϫ2 ΄ Ϫ5 0 0 0 2 1 0 0 Ϫ1 3 1 0 1 (a) 0 0 1 0 (c) 0 0 ΅ ΄ 1 0 0 0 1 0 5 3 0 ΅ ΅ ΄ 0 1 0 0 0 0 1 0 Ϫ1 2 3 0 ΅ 0 (b) 0 0 3 Ϫ2 4 1 1 0 (d) 0 0 The matrices shown in parts (b) and (d) are in reduced row-echelon form. The matrices listed below are not in row-echelon form. 2 Gaussian Elimination and Gauss-Jordan Elimination 19 It can be shown that every matrix is row-equivalent to a matrix in row-echelon form. For instance, in Example 4 you could change the matrix in part (e) to row-echelon form by 1 multiplying the second row in the matrix by 2 .

X ϩ 2͒͑x Ϫ 2͒2 x ϩ 2 x Ϫ 2 ͑x Ϫ 2͒2 29. 20 I6 16 volts R1 = 4 R3 = 4 R4 = 2 8 volts Chapter 1 32. 3x2 Ϫ 7x Ϫ 12 B C A ϩ ϩ ϭ ͑x ϩ 4͒͑x Ϫ 4͒2 x ϩ 4 x Ϫ 4 ͑x Ϫ 4͒2 In Exercises 33 and 34, find the values of x, y, and that satisfy the system of equations. Such systems arise in certain problems of calculus, and is called the Lagrange multiplier. ϩ ϭ0 2y ϩ ϭ0 xϩ y Ϫ4ϭ0 33. 2x 34. 2y ϩ 2 ϩ 2 ϭ 0 2x ϩ 1ϭ0 2x ϩ y Ϫ 100 ϭ 0 35. In Super Bowl XLI on February 4, 2007, the Indianapolis Colts beat the Chicago Bears by a score of 29 to 17.

For hand computations, fractions can sometimes be avoided by judiciously choosing the order in which elementary row operations are applied. REMARK : No matter which order you use, the reduced row-echelon form will be the same. The next example demonstrates how Gauss-Jordan elimination can be used to solve a system with an infinite number of solutions. EXAMPLE 8 A System with an Infinite Number of Solutions Solve the system of linear equations. 2x1 ϩ 4x 2 Ϫ 2x3 ϭ 0 ϭ1 3x1 ϩ 5x2 SOLUTION The augmented matrix of the system of linear equations is ΄3 2 4 5 Ϫ2 0 ΅ 0 .