Page 844 - Advanced_Engineering_Mathematics o'neil

824 Answers to Selected Problems

5. T is linear and
⎛ ⎞
1 0 0 −1 0
A T = ⎝ 0 1 −1 0 0 ⎠ .
0 0 0 1 1

T is onto but not one-to-one; A T has rank 3.
7. Not linear because of the sin(xy) term.
9. T is not linear (the image of a sum of two vectors has last component 2).

CHAPTER EIGHT DETERMINANTS
Section 8.1 Definition of the Determinant
n
1. |B|= α |A|
t
n
n
3. |A|= (−1) |A |= (−1) |A|.
n
Since (−1) =−1if n is odd, then |A|=−|A|,so |A|= 0.
Section 8.2 Evaluation of Determinants I
1. −22 3. −14 5. −2247 7. −122
9. 72
Section 8.3 Evaluation of Determinants II
1. 32 3. 3 5. −773 7. −152
9. 1693
11.

1 α α 2 1 α α 2
2
2
2

1 β β = 0 β − α β − α

2
2
2

1 γ γ 0 γ − α γ − α

1 α α 2
1 β + α
= (β − α)(γ − α) 0 1 β + α = (β − α)(γ − α)

1

0 1 γ + α γ + α,
leading to the value β − α)(γ − α)(γ − β).
Section 8.4 A Determinant Formula for A −1
1 6 1
1.
13 −1 2

1 −4 1
3.
5 1 1
⎛ ⎞
5 3 1
1
5. ⎝ −8 −24 24 ⎠
32 −2 −14 6
⎛ ⎞
−1 25 −21
1
7. ⎝ −8 −3 6 ⎠
29 −1 −4 8
210 −42 42 0
⎛ ⎞
1 ⎜ 899 −124 223
9. −135⎟
378 ⎝ 275 −64 109 −27 ⎠
−601 122 −131 81
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October 14, 2010 17:50 THM/NEIL Page-824 27410_25_Ans_p801-866