Page 242 - Advanced engineering mathematics

222 CHAPTER 7 Matrices and Linear Systems

Step 1. Find the general solution H of AX = O.
Step 2. Find any one solution U p of AX = B.
Step 3. The general solution AX = B is then H + U p .

We know how to carry out step (1). We will outline a procedure for step (2).
To find a particular solution U p , proceed as follows.
.
.
Step 1. Define the n × m + 1 augmented matrix [A.B] by adjoining the column matrix B as an
additional column to A. The augmented matrix contains the coefficients of the unknowns
of the system (in the first m columns), as well as the numbers on the right side of the
equations (elements of B).
.
.
Step 2. Reduce [A.B]. Since we reduce a matrix to obtain leading entries of 1 wherever possible
from upper left toward the lower right, this results eventually in a reduced matrix
. .
.
.
[A.B] R =[A R .C],
in which the first m columns are the reduced form of A, and the last column is whatever
.
.
results from B after the row operations used to reduce A have been applied to [A.B].
Solutions of the reduced system A R X = C are the same as solutions of the original
system AX = B because the operations performed on the coefficients of the unknowns
are also performed on the b j ’s.
.
.
Step 3. From [A R .C], read a particular solution U p . When added to the general solution H of the
associated homogeneous system, we have the general solution of AX = B.
We will look at some examples. Example 7.24 suggests how this augmented matrix
procedure tells us when the system has no solution.

EXAMPLE 7.23
We will solve the system

⎛ ⎞ ⎛ ⎞
−3 2 2 8
⎝ 1 4 −6 ⎠ X = ⎝ 1 ⎠ .
0 −2 2 −2
The first step is to reduce the augmented matrix

.
⎛ ⎞
−3 2 2 . . 8
. ⎜ ⎟
. ⎜ . . ⎟
1 4 −6 . 1 ⎟.
[A.B]= ⎜
⎝ ⎠
.
0 −2 2 . . −2
Carrying out the reduction procedure on this 3 × 4 augmented matrix, we obtain
.
⎛ ⎞
1 0 0 . . 0
. ⎜ ⎟ . .
. ⎜ . . ⎟ . .
[A.B] R = ⎜ 0 1 0 . 5/2 ⎟ =[A R .C]=[I 3 .C].
.
⎝ ⎠
0 0 1 . . 3/2
.
.
C is whatever results in the fourth column when we reduce A, the first three columns of [A.B]).

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October 14, 2010 14:23 THM/NEIL Page-222 27410_07_ch07_p187-246