Page 845 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 825

Section 8.5 Cramer’s Rule
1. x 1 =−11/47, x 2 =−100/47, 3. x 1 =−1/2, x 2 =−19/22, x 3 = 2/11
5. x 1 = 5/6, x 2 =−10/3, x 3 =−5/6 7. x 1 =−86, x 2 =−109/2, x 3 =−43/2, x 4 = 37/2
9. x 1 = 33/93, x 2 =−409/33, x 3 =−1/93, x 4 = 116/93

Section 8.6 The Matrix Tree Theorem
1.
2 0 −1 0 −1
⎛ ⎞
⎜ 0 2 −1 −1 0 ⎟
T = ⎜−1 −1 4 −1 −1⎟
⎜
⎟
⎝ 0 −1 −1 3 −1 ⎠
−1 0 −1 −1 3
and the number of spanning trees is 21.
3. 61 5. 61

CHAPTER NINE EIGENVALUES AND DIAGONALIZATION
Eigenvalues and Eigenvectors
2
1. p A (λ) = λ − 2λ − 5; eigenvalues and corresponding eigenvectors are
√ √ √ √
− 6
6
1 + 6, ;1 − 6, ;3.
2 2
The Gerschgorin circles are of radius 3 about (1,0) and radius 2 about (1,0).
3. p A (λ) = λ + 3λ − 10;
2

7 0
−5, ; 2,
−1 1
The Gerschgorin circle has radius 1, center (2,0).
2
5. p A (λ) = λ − 3λ + 14;
√ √

1 √ −1 + 47i 1 √ −1 − 47i
(3 + 47i), , (3 − 47i),
2 4 2 4
Gerschgorin circles have radius 6, center (1,0) and radius 2, center (2,0).
3
2
7. p A (λ) = λ − 5λ + 6λ,
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 2 0
0, ⎝ 1 ⎠ ; 2, ⎝ 1 ⎠ ; 3, ⎝ 2 ⎠
0 0 3
The Gerschgorin circle has radius 3, center (0,0).
3
9. p A (λ) = λ (λ + 3),
⎛ ⎞ ⎛ ⎞
1 1
0,0, ⎝ 0 ⎠ ;−3, ⎝ 0 ⎠
3 0
The Gerschgorin circle has radius 2, center (−3,0).
2
11. p A (λ) = (λ + 14)(λ − 2) ,
⎛ ⎞ ⎛ ⎞
−16 0
−14, ⎝ 0 ⎠ ;2,2, ⎝ 0 ⎠
1 1
The eigenvalue 2 of multiplicity 2 does not have two linearly independent eigenvectors. The Gerschgorin circles have
radius 1 and center (−14,0) and radius 1 and center (2,0).

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October 14, 2010 17:50 THM/NEIL Page-825 27410_25_Ans_p801-866