Page 282 - Advanced engineering mathematics

262 CHAPTER 8 Determinants

We find that |A|= 13, so this system has a unique solution. By Cramer’s rule,

1 −3 −4

1 117
x 1 = 14 1 −3 =− =−9,

13 13
5 1 −3

1 1 −4

1 10
x 2 = −114 −3 =− ,

13 13
0 5 −3

1 −3 1

1 25
x 3 = −1 1 14 =− .

13 13
0 1 5
SECTION 8.5 PROBLEMS
In each of Problems 1 through 10, solve the system using x 1 − 3x 2 + x 3 − 4x 5 = 0
Cramer’s rule, or show that the rule does not apply because −2x 1 + x 3 − 2x 5 = 4
the matrix of coefficients is singular. x 3 − x 4 − x 5 = 8
7. 2x 1 − 4x 2 + x 3 − x 4 = 6
1. 15x 1 − 4x 2 = 5 x 2 − 3x 3 = 10
8x 1 + x 2 =−4 x 1 − 4x 3 = 0
x 2 − x 3 + 2x 4 = 4
2. x 1 + 4x 2 = 3
x 1 + x 2 = 0 8. 2x 1 − 3x 2 + x 4 = 2
3. 8x 1 − 4x 2 + 3x 3 = 0 x 2 − x 3 + x 4 = 2
x 1 + 5x 2 − x 3 =−5 x 3 − 2x 4 = 5
−2x 1 + 6x 2 + x 3 =−4 x 1 − 3x 2 + 4x 3 = 0
4. 5x 1 − 6x 2 + x 3 = 4 9. 14x 1 − 3x 3 = 5
−x 1 + 3x 2 − 4x 3 = 5 2x 1 − 4x 3 + x 4 = 2
2x 1 + 3x 2 + x 3 =−8 x 1 − x 2 + x 3 − 3x 4 = 1
5. x 1 + x 2 − 3x 3 = 0 x 3 − 4x 4 =−5
x 2 − 4x 3 = 0 10. x 2 − 4x 4 = 18
x 1 − x 2 − x 3 = 5 x 1 − x 2 + 3x 3 =−1
6. 6x 1 + 4x 2 − x 3 + 3x 4 − x 5 = 7 x 1 + x 2 − 3x 3 + x 4 = 5
x 1 − 4x 2 + x 5 =−5 x 2 + 3x 4 = 0

8.6 The Matrix Tree Theorem

In 1847, G.R. Kirchhoff published a classic paper in which he derived many of the electrical
circuit laws that bear his name, including the matrix tree theorem we will now discuss.
Figure 8.1 shows a typical electrical circuit. The underlying geometry of the circuit if shown
in Figure 8.2. Such a diagram of points and interconnecting lines is called a graph, and was seen
in the context of atoms moving through crystals in Section 7.1.3. A labeled graph has symbols
attached to the points.
Some of Kirchhoff’s results depend on geometric properties of the circuit’s underlying
graph. One such property is the arrangement of the closed loops. Another is the number of

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October 14, 2010 14:26 THM/NEIL Page-262 27410_08_ch08_p247-266