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

March 31, 2013 16-132x 1 + 3x 2 + x 3 = 2x 1 − x 2 − x 3 = 13x 1 + 2x 2 = 3Notice that this system only differs from that in the preceding exampleby the a change of sign in the multiplier of x 2 in the third equation. Thesolutions, however, will change considerably. In Example 2, there was aunique solution. In this example, it will be seen that there are infinitelymany solutions.We begin with the augmented matrix, and proceed to use the row operations.Instead of writing each changed matrix, we combine several steps.2 3 1 21 -1 -1 13 2 0 3−2R 2 + R 1 followed by −3R 2 + R 30 5 3 01 -1 -1 10 5 3 0Now, this reduces to the two equationsx 1 = 1 + x 2 + x 35x 2 + 3x 3 = 0x 2 = −(3/5)x 3x 1 = 1 − (3/5)x 3 + x 3x 1 = 1 + (2/5)x 3So we see that solutions have the form⎡⎤ ⎡ ⎤ ⎡1 + (2/5)x 3 1⎢⎥ ⎢ ⎥ ⎢⎣ −(3/5)x 3 ⎦ = ⎣ 0 ⎦ + x 3 ⎣x 3 02/5−3/51⎤⎥⎦