17.4 The solution of linear equations 487
EXAMPLE 17.8Find the values of λfor which the following system of equations
has nonzero solution:
− 2 x+ y+ z= 1 λx
− 11 x 1 + 14 y 1 + 15 z= 1 λy
−x+ y = 1 λz
The equations can be written
(− 21 − 1 λ)x 1 + y + z = 10
− 11 x + 1 (4 1 − 1 λ)y 1 + 5 z = 10
−x + y + 1 (−λ)z 1 = 10
and have nonzero solution when
Therefore
= 1 (− 21 − 1 λ)[−λ(4 1 − 1 λ) 1 − 1 5] 1 − 1 [11λ 1 + 1 5] 1 + 1 [− 111 + 1 (4 1 − 1 λ)]
= 1 −λ
31 + 12 λ
21 + 1 λ 1 − 121 = 1 −(λ 1 − 1 1)(λ 1 + 1 1)(λ 1 − 1 2)
andD 1 = 10 whenλ 1 = 11 , − 1 , and 2.
For each of the nroots of the secular determinant there exists a solution of the secular
equations (17.38).
EXAMPLE 17.9Solve the Hückel molecular-orbital problem for the allyl radical
CH
2CH CH
2in terms of the Hückel parameters αand β:
(1) (α 1 − 1 E)c
11 + βc
2= 10
(2) βc
1- 1 (α 1 − 1 E)c
21 + βc
3= 10
(3) βc
2- 1 (α 1 − 1 E)c
3= 10
D=− −
−
−
−
−
−−
−−
−
()2
45
1
11 5
1
11 4
11
λ
λ
λλ
λ
D=
−−
−−
−−
=
211
11 4 5
11
0
λ
λ
λ