Page 243 - Advanced engineering mathematics
P. 243
7.6 Nonhomogeneous Systems 223
.
.
This reduced augmented matrix [A R .C] represents the reduced system I 3 X=C, which is the
system
x 1 = 0
x 2 = 5/2
x 3 = 3/2.
From this, we directly read a solution of the reduced nonhomogeneous system. In this example,
this solution is unique by Corollary 7.6 because the associated homogeneous system has only the
trivial solution (A R = I 3 ).
Consistent with treating the system as a matrix equation, we usually write the solution in
terms of column matrices. In this example,
⎛ ⎞
0
.
X = 5/2 ⎠
⎝
3/2
EXAMPLE 7.24
We have seen that the system
2x 1 − 3x 2 = 6
4x 1 − 6x 2 = 8
has no solution. We will see how this conclusion reveals itself when we work with the augmented
matrix, which is
.
⎛ ⎞
. . 2 3 . . 6
[A.B]= ⎝ ⎠ .
.
4 −6 . . 8
Reduce this matrix to obtain
.
⎛ ⎞
. . 1 −3/2 . . 2
.
.
[A.B] R =[A R .C]= ⎝ ⎠ .
.
0 0 . . −4
The second equation of the reduced system is
0x 1 + 0x 2 =−4
which can have no solution.
In this example, the augmented matrix has rank 2, while the matrix of the homogeneous
.
.
system has rank 1. In general, whenever the rank of A is less than the rank of [A.B], then A R will
have at least one row of zeros, while the corresponding row in the reduced augmented matrix
.
.
[A R .C] has a nonzero element in this row in the C column. This corresponds to an equation of
the form
0x 1 + 0x 2 + ··· + 0x m = c j = 0
and this has no solution for the x i ’s. In this case the system is inconsistent.
We will record this important observation.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
October 14, 2010 14:23 THM/NEIL Page-223 27410_07_ch07_p187-246
