These are examples done in class with MATLAB on Thursday, 9/9/04. The example below is the augmented matrix for the system we did on Tuesday, 9/7/04. The lines below enter the augmented matrix, which MATLAB then echoes. aug=[1 -2 1 -5; 2 -3 4 -12; 3 1 2 1] aug = 1 -2 1 -5 2 -3 4 -12 3 1 2 1 The next command runs MATLAB's rrefmovie (rref stands for reduced echelon form). It is a package that helps illustrate row reducing a matrix. rrefmovie(aug) Original matrix A = 1 -2 1 -5 2 -3 4 -12 3 1 2 1 Press any key to continue. . . swap rows 1 and 3 A = 3 1 2 1 2 -3 4 -12 1 -2 1 -5 Press any key to continue. . . pivot = A(1,1) A = 1 1/3 2/3 1/3 2 -3 4 -12 1 -2 1 -5 Press any key to continue. . . eliminate in column 1 A = 1 1/3 2/3 1/3 2 -3 4 -12 1 -2 1 -5 Press any key to continue. . . A = 1 1/3 2/3 1/3 0 -11/3 8/3 -38/3 1 -2 1 -5 A = 1 1/3 2/3 1/3 0 -11/3 8/3 -38/3 0 -7/3 1/3 -16/3 Press any key to continue. . . pivot = A(2,2) A = 1 1/3 2/3 1/3 0 1 -8/11 38/11 0 -7/3 1/3 -16/3 Press any key to continue. . . eliminate in column 2 A = 1 1/3 2/3 1/3 0 1 -8/11 38/11 0 -7/3 1/3 -16/3 Press any key to continue. . . A = 1 0 10/11 -9/11 0 1 -8/11 38/11 0 -7/3 1/3 -16/3 A = 1 0 10/11 -9/11 0 1 -8/11 38/11 0 0 -15/11 30/11 Press any key to continue. . . pivot = A(3,3) A = 1 0 10/11 -9/11 0 1 -8/11 38/11 0 0 1 -2 Press any key to continue. . . eliminate in column 3 A = 1 0 10/11 -9/11 0 1 -8/11 38/11 0 0 1 -2 Press any key to continue. . . A = 1 0 0 1 0 1 -8/11 38/11 0 0 1 -2 A = 1 0 0 1 0 1 0 2 0 0 1 -2 Press any key to continue. . . The last "press any key ..." gives back the command promt. Of course, the final matrix above, which is the row reduced form of the the original augmented matrix, allows us to simply read off the solution to the original problem. In this case, x1 = 1, x2 = 2, and x3 = -1. We now enter the augmented matrix for a new system, one with 2 equations and 4 unknowns. aug=[1 -2 1 2 -1; -1 2 0 -3 2] aug = 1 -2 1 2 -1 -1 2 0 -3 2 rrefmovie(aug) Original matrix A = 1 -2 1 2 -1 -1 2 0 -3 2 Press any key to continue. . . pivot = A(1,1) A = 1 -2 1 2 -1 -1 2 0 -3 2 Press any key to continue. . . eliminate in column 1 A = 1 -2 1 2 -1 -1 2 0 -3 2 Press any key to continue. . . A = 1 -2 1 2 -1 0 0 1 -1 1 Press any key to continue. . . column 2 is negligible A = 1 -2 1 2 -1 0 0 1 -1 1 Press any key to continue. . . pivot = A(2,3) A = 1 -2 1 2 -1 0 0 1 -1 1 Press any key to continue. . . eliminate in column 3 A = 1 -2 1 2 -1 0 0 1 -1 1 Press any key to continue. . . A = 1 -2 0 3 -2 0 0 1 -1 1 Press any key to continue. . . (This reutrns the prompt.) Our final example is a system that is inconsistent - i.e., it has no solution. As before, we enter its augmented matrix, and then row reduce that martix. aug=[1 1 1 2; 1 -1 2 0; 2 0 3 9] aug = 1 1 1 2 1 -1 2 0 2 0 3 9 rrefmovie(aug) Original matrix A = 1 1 1 2 1 -1 2 0 2 0 3 9 Press any key to continue. . . swap rows 1 and 3 A = 2 0 3 9 1 -1 2 0 1 1 1 2 Press any key to continue. . . pivot = A(1,1) A = 1 0 3/2 9/2 1 -1 2 0 1 1 1 2 Press any key to continue. . . eliminate in column 1 A = 1 0 3/2 9/2 1 -1 2 0 1 1 1 2 Press any key to continue. . . A = 1 0 3/2 9/2 0 -1 1/2 -9/2 1 1 1 2 A = 1 0 3/2 9/2 0 -1 1/2 -9/2 0 1 -1/2 -5/2 Press any key to continue. . . pivot = A(2,2) A = 1 0 3/2 9/2 0 1 -1/2 9/2 0 1 -1/2 -5/2 Press any key to continue. . . eliminate in column 2 A = 1 0 3/2 9/2 0 1 -1/2 9/2 0 1 -1/2 -5/2 Press any key to continue. . . A = 1 0 3/2 9/2 0 1 -1/2 9/2 0 1 -1/2 -5/2 A = 1 0 3/2 9/2 0 1 -1/2 9/2 0 0 0 -7 Press any key to continue. . . column 3 is negligible A = 1 0 3/2 9/2 0 1 -1/2 9/2 0 0 0 -7 At this point in the process, we can see that the system has no solution. The reason is the last row gives us the equation, 0·x1 + 0·x2 + 0·x3 = -7, which implies 0 = -7. Since this is impossible, the system has no solution. We will let MATLAB finish row reducing the matrix. Press any key to continue. . . pivot = A(3,4) A = 1 0 3/2 9/2 0 1 -1/2 9/2 0 0 0 1 Press any key to continue. . . eliminate in column 4 A = 1 0 3/2 9/2 0 1 -1/2 9/2 0 0 0 1 Press any key to continue. . . A = 1 0 3/2 0 0 1 -1/2 9/2 0 0 0 1 A = 1 0 3/2 0 0 1 -1/2 0 0 0 0 1 Press any key to continue. . . diary off