Page 244 - Advanced engineering mathematics

224 CHAPTER 7 Matrices and Linear Systems

THEOREM 7.14
.
.
The nonhomogeneous system AX = B is consistent if and only if A and [A.B] have the same
rank.

EXAMPLE 7.25
We will solve the system
x 1 − x 2 + 2x 3 = 3
−4x 1 + x 2 + 7x 3 =−5
−2x 1 − x 2 + 11x 3 = 14.
The augmented matrix is
.
⎛ ⎞
1 −1 2 . . 3
. ⎜ ⎟
. ⎜ . ⎟
−4 1 7 . . −5 ⎟.
[A.B]= ⎜
⎝ ⎠
.
−2 −1 11 . . 14
Reduce this augmented matrix to obtain
⎛ ⎞
.
1 0 −3 . . 0
. . ⎜ ⎟
. . ⎜ . . ⎟.
⎟
0 1 −5 . 0
[A.B] R =[A R .C]= ⎜
⎝ ⎠
.
0 0 0 . . 1
.
.
A has rank 2, because its reduced matrix has two nonzero rows. But [A.B] has rank 3 because its
reduced form has three nonzero rows. Therefore, this system is inconsistent. We can also observe
from the reduced system that the last equation is
0x 1 + 0x 2 + 0x 3 = 1
with no solution.
EXAMPLE 7.26
Solve the system
x 1 − x 2 + 2x 4 + x 5 + 6x 6 =−3
x 2 + x 3 + 3x 4 + 2x 5 + 4x 6 = 1
x 1 − 4x 2 + 3x 3 + x 4 + 2x 6 = 0.
The augmented matrix is
⎛ ⎞
.
1 0 −1216 . . −3
. ⎜ ⎟
. ⎜ . ⎟
0 1 1 3 2 4 . . 1 ⎟.
[A.B]= ⎜
⎝ ⎠
.
1 −4 3 1 0 2 . . 0
Reduce this to obtain
.
⎛ ⎞
10027/8 15/8 60/8 . . −17/8
. . ⎜ ⎟
. . ⎜ . . ⎟
[A.B] R =[A R .C]⎜ 01013/8 9/8 20/8 . 1/8 ⎟.
.
⎝ ⎠
00111/8 7/8 12/8 . . 7/8
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October 14, 2010 14:23 THM/NEIL Page-224 27410_07_ch07_p187-246