Page 245 - Advanced engineering mathematics

7.6 Nonhomogeneous Systems 225

. .
.
.
The first six columns are A R , and we read from [A.B] R that both A and [A.B] have rank 3,
so the system is consistent. From the reduced augmented matrix, we see immediately that
27 15 60 17
x 1 + x 4 + x 5 + x 6 =−
8 8 8 8
13 9 20 1
x 2 + x 4 + x 5 + x 6 =
8 8 8 8
11 7 12 7
x 3 + x 4 + x 5 + x 6 = .
8 8 8 8
From these we have
27 15 60 17
x 1 =− x 4 − x 5 − x 6 −
8 8 8 8
13 9 20 1
x 2 =− x 4 − x 5 − x 6 +
8 8 8 8
11 7 12 7
x 3 =− x 4 − x 5 − x 6 + .
8 8 8 8
We could have gone directly to these equations without the intermediate step. These equations
actually give the general solution, with x 1 , x 2 , and x 3 in terms of the arbitrary constants x 4 , x 5 ,
and x 6 . The solution is
27 15 60 17
⎛ ⎞
− x 4 − x 5 − x 6 −
8 8 8 8
13 9 20 1 ⎟
⎜ − x 4 − x 5 − 8 x 6 + 8 ⎟
⎜
8
8
⎜ 11 7 12 7 ⎟
⎜ − x 4 − x 5 − x 6 + ⎟
8 8 8 8 ⎟.
X = ⎜
x 4
⎜ ⎟
⎜ ⎟
⎝ ⎠
x 5
x 6
To write this in a more revealing way, let x 4 = 8α, x 5 = 8β, and x 6 = 8γ to write
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
−27 −15 −60 −17/8
−13 −9 −20 1/8
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
−11 −7 −12 7/8
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
X = α ⎜ ⎟ + β ⎜ ⎟ + γ ⎜ ⎟ + ⎜ ⎟ = H + U p
8 0 0 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
0
⎝ 0 ⎠ ⎝ 8 ⎠ ⎝ 0 ⎠ ⎝ ⎠
0 0 8 0
with H as the general solution of AX = O and U p as a particular solution of AX = C.
EXAMPLE 7.27
The system
⎛ ⎞ ⎛ ⎞
2 1 −11 −6
⎝ −51 X = .
9 ⎠ ⎝ 12 ⎠
1 1 14 −5
has the augmented matrix
.
⎛ ⎞
2 1 −11 . . −6
. ⎜ ⎟
. ⎜ . . ⎟
−51 9 . 12 ⎟,
[A.B]= ⎜
⎝ ⎠
.
1 1 14 . . −5
and we reduce this to
.
⎛ ⎞
100 . . −86/31
. ⎜ ⎟
. ⎜ . . ⎟
[A.B] R = ⎜ 010 . −191/155 ⎟.
⎝ ⎠
.
001 . . −11/155
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October 14, 2010 14:23 THM/NEIL Page-225 27410_07_ch07_p187-246