Page 841 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 821

7.
⎛ ⎞ ⎛ ⎞
−5/6 −5/9
⎜−2/3⎟ ⎜−10/9⎟
⎟ ⎜ ⎟
⎜
⎜−8/3⎟ ⎜−13/9⎟
α ⎜ ⎟ + β ⎜ ⎟
⎜−2/3⎟ ⎜ −1/9 ⎟
⎝ 1 ⎠ ⎝ 0 ⎠
0 1
The solution space has dimension 2.
9.
5/14 x 1
⎛ ⎞ ⎛ ⎞
⎜11/7⎟ ⎜x 2⎟
6/7 ⎠ = ⎝ ⎠
α ⎝
x 4
1 x 5
The solution space has dimension 1.
11.
1 −2 0
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎜1⎟ ⎜−3/2⎟ ⎜1/2⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜0⎟ ⎜ 2/3 ⎟ ⎜−3⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
α ⎜1⎟ + β ⎜−4/3⎟ + γ ⎜ 0 ⎟
⎟
⎜ ⎟ ⎜ ⎟ ⎜
⎜ 1 ⎟ ⎜ 0 ⎟ ⎜ 0 ⎟
0 1 0
⎝ ⎠ ⎝ ⎠ ⎝ ⎠
0 0 1
The solution space has dimension 3.
13. Yes, if m − rank (A)> 0. For example, the system
x 1 + 3x 2 = 0,2x 1 + 6x 2 = 0,3x 1 + 9x 3 = 0
has the solution x 1 =−3x 3 , x 2 = x 3 and the solution space has dimension 1.

Section 7.6 Nonhomogeneous Systems
1. Unique solution
⎛ ⎞
1
⎝ 1/2 ⎠
4
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 −17/2 9/2
⎜ 1 ⎟ ⎜ 0 ⎟ ⎜ −6 ⎟ ⎜ 3 ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜3/2⎟ ⎜1/2⎟ ⎜−51/4⎟ ⎜25/4⎟
0
3. α ⎜ ⎟ + β ⎜ ⎟ + γ ⎜ ⎟ + ⎜ ⎟
⎜ 1 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟
0 1 0 0
⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
0 0 1 0
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
2 −2 −4
⎜ 2 ⎟ ⎜ −1 ⎟ ⎜ −4 ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ 7 ⎟ ⎜−9/2⎟ ⎜ −38 ⎟
5. α ⎜ ⎟ + β ⎜ ⎟ + ⎜ ⎟
⎜3/2⎟ ⎜−3/4⎟ ⎜−11/2⎟
⎝ 1 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠
0 1 0
−1/2 −3/4 9/8
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎜ −1 ⎟ ⎜ 1 ⎟ ⎜ 2 ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
7. α ⎜ 3 ⎟ + β ⎜ −2 ⎟ + ⎜ 0 ⎟
⎝ 1 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠
0 1 0
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October 14, 2010 17:50 THM/NEIL Page-821 27410_25_Ans_p801-866