Page 47 - A Course in Linear Algebra with Applications

2.1: Gaussian Elimination 31

X i - x 2 +a: 3 + x 4 = 2
2x 2 - 2x 4 = 1
4x 2 - 4x 4 = -1

Next multiply equation 2 of this new system by \, an opera-
tion which is denoted by \ (2), to get

+
xi - - x 2 +£3 + x 4 = = 2
+x 3
x 2
x 4
1.
— X4 =
2
X 2
4x 2 - 4x 4 = -1
Finally, eliminate x 2 from equation 3 by performing the op-
eration (3) — 4(2), that is, subtract 4 times equation 2 from
equation 3; this yields the linear system
Xi - x 2 + x 3 + x 4 = 2
X2 — X4 1
—
2
0 = -3
Of course the third equation is false, so the original linear
system has no solutions, that is, it is inconsistent.

Example 2.1.2

X\ + 4x 2 + 2X 3 = -2
+ 3x 3 = 32
2xi - 8x 2
+ x = 1
X 2 3
Add two times equation 1 to equation 2, that is, perform the
operation (2) + 2(1), to get

xi + 4x 2 + 2x 3 = - 2
= 28
7x 3
x 2 + x 3 — 1