Page 842 - Advanced_Engineering_Mathematics o'neil

822 Answers to Selected Problems

−1 1 −3/14 −1 1/14 −29/7
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 0 0
⎜ 1 ⎟ ⎜0⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ 0 ⎟ ⎜0⎟ ⎜ 3/14 ⎟ ⎜ 0 ⎟ ⎜−1/14⎟ ⎜ 1/7 ⎟
0 0 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
9. α ⎜ 0 ⎟ + β ⎜1⎟ + γ ⎜ ⎟ + δ ⎜ 0 ⎟ + ⎜ ⎟ + ⎜ ⎟
1 0 0
⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜ 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟
0 0 0 ⎠ ⎝ −1 ⎠ ⎝ 0 ⎠ ⎝ 0 ⎠
⎝ ⎠ ⎝ ⎠ ⎝
0 0 0 0 1 0
−19/15 22/15
⎛ ⎞ ⎛ ⎞
3 −5
⎜ ⎟ ⎜ ⎟
67/15 ⎠ + ⎝ −121/15 ⎠
11. α ⎝
1 0
13. Unique solution
⎛ ⎞
16/57
X = ⎝ 99/57 ⎠
23/57
15. If AX = B is consistent, then there is a solution C.Since AC is a linear combination of the columns of A (see Section
7.1.1), so B is in the column space of A.Conversely, if B is in the column space of A,then B is a linear combination
a 1 A 1 + ··· + a m A m
of the columns of A,andthen AC = B,where C is the column matrix of the coefficients a 1 ,··· ,a m .
Section 7.7 Matrix Inverses

1 −1 2
1.
5 2 1
1 −2 2 1 3 −2
3. 5.
12 1 5 12 −3 6
⎛ ⎞
−6 11 2
1
7. ⎝ 3 10 −1 ⎠
31 1 −7 10
⎛ ⎞
6 −6 0
1
9. − ⎝ −3 −9 2 ⎠
12 3 −3 −2
−1 −1 8 4 1 −23
⎛ ⎞⎛ ⎞ ⎛ ⎞
1 ⎜−9 2 −5
−1
11. X = A B = 14⎟⎜ 2 ⎟ 1 ⎜−75⎟
⎠ =
11 ⎝ 2 2 −5 3 ⎠⎝ 0 11 ⎝ −9 ⎠
3 3 −2 −1 −5 14
⎛ ⎞
22
1
13. ⎝ 27 ⎠
7
30
⎛ ⎞
−21
1
15. ⎝ 14 ⎠
5
0
Section 7.8 The Method of Least Squares and Data Fitting

13/5
∗
1. X =
7/5

−2
∗
3. X =
4
⎛ ⎞ ⎛ ⎞
7/3 −2
∗
5. X = α ⎝ 1 ⎠ + ⎝ 0 ⎠
5/3 −1
7. y = 3.88x + 0.16 9. y = 4.164x − 9.267
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October 14, 2010 17:50 THM/NEIL Page-822 27410_25_Ans_p801-866