16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-11⎡⎢⎣∣ ⎤a 11 a 12 . . . a 1n ∣∣∣∣∣∣∣∣∣ b 1a 21 a 22 . . . a 2n b 2⎥.. ⎦a m1 a m2 . . . a mn b mThis is called the augmented matrix of the system.We manipulate this matrix with the operations above to put the partcorresponding to A in a better form so that we can read off the solutions.Example 1. Consider the system2x 1 + 3x 2 = 4x 1 − x 2 = 6Of course, we can solve this by Cramer’s rule. We first do this. Afterward, weuse the method of row operations. This latter method provides a systematicway to handle general linear systems of equations.Using Cramer’s rule, we first write the system in terms of matrices:[2 31 −1] [ ] [ ]x1 4=x 2 6(7)Then, we getx 1 =([ ])4 3det6 −1([ ]) = 22/5, x 2 =2 3det1 −1([ ])2 4det1 6([ ]) = −8/52 3det1 −1Next, let’s do this with row operations. We denote row i by R i .Write the augmented matrix:2 3 41 -1 60 5 -81 -1 6