The Chemistry Maths Book, Second Edition

488 Chapter 17Determinants

The equations have nonzero solution when the determinant of the coefficients ofc

1

,

c

2

,andc

3

is zero:

The roots areE

1

1 = 1 α,and

The corresponding solutions of the equations are obtained by replacing Ein the

secular equations by each root in turn.

ForE 1 = 1 E

1

1 = 1 α:(1)βc

2

1 = 10 c

2

1 = 10

(2) βc

1

1 + 1 βc

3

1 = 10 c

1

1 = 1 −c

3

(3) βc

2

1 = 10 c

2

1 = 10

We see that equations (1) and (3) are identical, so that only two of the three equations

are independent. We solve forc

1

andc

2

in terms of (arbitrary)c

3

. Similarly,

Settingc

3

1 = 11 for convenience, the three solutions of the secular problem are therefore

Ec

1

c

2

c

3

α − 101

11

11

0 Exercises 19–21

17.5 Properties of determinants

The following are the more important general properties of determinants.

1. Transposition

Because the same value of a determinant is obtained by expansion along any row

orcolumn,

the value of a determinant is unchanged if its rows and columns

are interchanged:

αβ− 2 − 2

αβ+ 2 2

cc c

12 3

=− 2 =

→

()12 0

12

ββcc+=

cc

23

=− 2

→

()320

23

EE==− ββcc+=

3

αβ 2 :

cc c

12 3

== 2

→

()12 0

12

−+=ββcc

cc

23

= 2

→

()320

23

EE==+ ββcc−=

2

αβ 2 :

→

→

→

E

3

E =−αβ 2.

2

=+αβ 2

αβ

βα β

βα

αα β

−

−

−

=− − −













=

E

E

E

EE

0

0

20

22

()()