The Chemistry Maths Book, Second Edition

17.1 Concepts 475

Whena

1

b

2

1 − 1 b

1

a

2

is not zero, the required (unique) values of xand yare

(17.4)

The quantity in the denominators of (17.4) is a property of the coefficients in

equations (17.1), and is written in the form

(17.5)

The symbol on the left is called a determinant;

2

the expression on the right side

defines its value.

The solution (17.4) of the system of equations (17.1) can now be written as

(17.6)

where

(17.7)

EXAMPLE 17.1Use determinants to solve the equations

2 x 1 − 13 y 1 = 15

x 1 + 15 y 1 = 19

We have

Thereforex 1 = 1 D

1

2 D 1 = 14 andy 1 = 1 D

2

2 D 1 = 11.

0 Exercises 1– 4

DD

12

53

95

55 3 952

25

19

= 295113

−

=×−− ×= ,() = = ×−×=

D=

−

=×−−×=

23

15

25 3 113()

D

ab

ab

D

cb

cb

D

ac

ac

=,=,=

1

1

22

1

1

1

22

2

1

1

22

x

D

D

y

D

D

=, =

12

ab

ab

ab ba

1

1

22

12 12

=−

x

cb bc

ab ba

y

ac ca

ab ba

=

−

−

,=

−

−

12 12

12 12

12 12

12 12

2

The name determinant was coined by Cauchy in 1812 in the first of a long series of papers on the subject

of a class of alternating symmetric functions such asa

1

b

2

1 − 1 b

1

a

2

. In the general case, he arranged then

2

different

quantities in a square array, and used the abbreviation (a

1,n

) to represent the array with which a determinant is

associated. Cauchy used determinants for problems in geometry and in physics, and for the quantity now called

the Jacobian. He introduced the concept of minors and the expansion along any row or column.