896 CHAPTER 9 Systems and Matrices Special Systems The next two examples show how to recognize systems with no solutions or systems with infinitely many solutions when solving such systems using row transformations. £1 0 0 0 1 0 0 0 1 3 1 2 -1§ Final Matrix The linear system associated with this final matrix is x = 1 y = 2 z = -1. The solution set is 511, 2, -126. Check the solution in the original system. S Now Try Exercise 31. The TI-84 Plus graphing calculator is able to perform row operations. See Figure 7(a). The screen in Figure 7(b) shows typical entries for the matrix in the second step of the solution in Example 2. The entire Gauss-Jordan method can be carried out in one step with the rref (reduced row-echelon form) command, as shown in Figure 7(c). (If some of the numbers in Figure 7(c) had been decimals, they could have been converted to fractions with the Frac command from the MATH menu.) This is the matrix after column 1 has been transformed. (b) This screen shows the final matrix in the algebraic solution, found by using the rref command. (c) This menu shows various options for matrix row transformations in choices C–F. (a) Figure 7 7 EXAMPLE 3 Solving an Inconsistent System Use the Gauss-Jordan method to solve the system. x + y = 2 2x + 2y = 5 SOLUTION c 1 2 1 2 2 2 5d Write the augmented matrix. c 1 0 1 0 2 2 1d -2R1 + R2 The next step would be to introduce 1 in the second row, second column. Because of the 0 there, this is impossible. The second row corresponds to 0x + 0y = 1 which is false for all pairs of x and y, so the system has no solution. The system is inconsistent, and the solution set is ∅. S Now Try Exercise 27. 0 0
RkJQdWJsaXNoZXIy NjM5ODQ=