The Chemistry Maths Book, Second Edition

494 Chapter 17Determinants

(17.51)

Every determinant can be reduced to triangular form by means of a systematic

application of Property 7 in Section 17.5. The method is an example of the elimination

methods discussed in Chapter 20, and is illustrated in Example 17.17 for a third-order

determinant.

EXAMPLE 17.17Example of reduction to triangular form.

0 Exercises 26–27

17.7 Alternating functions

A functionf(x

1

, x

2

, x

3

, =, x

n

)of nvariables is called an alternating function, or

totally antisymmetric, if the interchange of any two of the variables has the effect of

multiplying the value of the function by (–1). For the interchange ofx

1

andx

2

,

f(x

2

, x

1

, x

3

, =, x

n

) 1 = 1 −f(x

1

, x

2

, x

3

, =, x

n

) (17.52)

=−−=×−×=−

123

044

001

1414()

subtract()row from row

1

4

× 23

=−−

−

123

044

010

=−− subtract()21 3×row from row

12 3

044

236

subtract row from row()31 2×

12 3

32 5

23 6

aaaa a

aaa a

aa a

n

n

n

11

12 13 14

1

22

23 24

2

33 34 3

0

00

0







000

0000

44 4

11 22 33

aa

a

aa a a

n

nn

nn





=××××