The Chemistry Maths Book, Second Edition

17 Determinants

17.1 Concepts

Many problems in the physical sciences, in engineering, and in statistics give rise to

systems of simultaneous linear equations. The methods of elementary algebra are

adequate when the number of such equations is small; two or three, as discussed

in Section 2.8. In some cases however the number of equations can be large, and

alternative methods are then required both for the numerical solution of the large

‘linear systems’, and for the formulation and theoretical analysis of the problems that

give rise to them. Some of the practical methods of solution, the ‘numerical methods’,

are discussed in Chapter 20. The branch of mathematics concerned with the theory of

linear systems is matrix algebra, the subject of Chapters 18 and 19, but several of the

more important and useful results in the theory of linear equations can be derived

independently from quantities called determinants. The theory of determinants is

discussed in this chapter as a separate topic, partly in preparation for the more general

matrix algebra of Chapters 18 and 19, and partly because determinants have certain

symmetry properties that have made them an important tool in quantum mechanics.

They are used in quantum chemistry to construct electronic wave functions that are

consistent with the requirements of the Pauli Exclusion Principle.

The concept of determinants has its origin in the solution of simultaneous linear

equations.

1

We consider the pair of equations

(1) a

1

x 1 + 1 b

1

y 1 = 1 c

1

(17.1)

(2) a

2

x 1 + 1 b

2

y 1 = 1 c

2

wherea

1

, b

1

, c

1

, a

2

, b

2

, andc

2

are constants. The equations are linear in the ‘unknowns’

xand y, and can be solved by the elementary methods of algebra. Thus, to solve for x,

we multiply equation (1) byb

2

and equation (2) byb

1

to give

(1′) a

1

b

2

x 1 + 1 b

1

b

2

y 1 = 1 c

1

b

2

(2′) b

1

a

2

x 1 + 1 b

1

b

2

y 1 = 1 b

1

c

2

so that, subtracting (2′) from (1′),

(a

1

b

2

1 − 1 b

1

a

2

)x 1 = 1 c

1

b

2

1 − 1 b

1

c

2

(17.2)

Similarly for y,

(a

1

b

2

1 − 1 b

1

a

2

)y 1 = 1 a

1

c

2

1 − 1 c

1

a

2

(17.3)

1

The earliest descriptions of the method of solving sets of linear equations by determinants, known as

Cramer’s method, were by the Japanese Seki Kowa (1642–1708) in a manuscript of 1683, and by Leibniz in a letter

to l’Hôpital in 1693 (published in 1850) in which he also gave the condition for the consistency of the equations.

The first published account appeared in MacLaurin’s Treatise of algebra(posthumously in 1748).