The Chemistry Maths Book, Second Edition

476 Chapter 17Determinants

The determinant (17.5) is a property of a square array of 41 = 12

2

elements, the

coefficients of xand yin the system of equations (17.1). It is a determinant of order2.

In the general case, a determinant of order nis a property of a square array ofn

2

elements, and is written

(17.8)

In this notation, the elementa

ij

lies in the ith row and jth column of the array; for

example,a

23

lies in the second row and third column. The determinant is sometimes

denoted by the symbol|a

ij

|. Determinants of order narise from the consideration of

systems of nlinear equations in nunknowns.

17.2 Determinants of order 3

The system of three linear equations in three unknowns,x

1

,x

2

andx

3

,

(1) a

11

x

1

1 + 1 a

12

x

2

1 + 1 a

13

x

3

1 = 1 b

1

(2) a

21

x

1

1 + 1 a

22

x

2

1 + 1 a

23

x

3

1 = 1 b

2

(17.9)

(3) a

31

x

1

1 + 1 a

32

x

2

1 + 1 a

33

x

3

1 = 1 b

3

can be solved forx

1

by multiplying

equation (1) by a

22

a

33

1 − 1 a

23

a

32

1 =

(2) by −(a

12

a

33

1 − 1 a

13

a

32

) 1 = 1 −

(3) by a

12

a

23

1 − 1 a

13

a

22

1 =

and adding the three results. The coefficient ofx

1

in the resulting expression defines

the determinant of order 3:

(17.10)

=−+a

aa

aa

a

aa

aa

a

aa

a

11

22

23

32 33

21

12

13

32 33

31

12

13

222 23

a

D

aaa

aaa

aaa

=

11

12 13

21 22 23

31 32 33

aa

aa

12

13

22 23

aa

aa

12 13

32 33

aa

aa

22 23

32 33

aaa a

aaa a

aaa a

n

n

n

11

12 13

1

21 22 23

2

31 32 33

3







aaaa a

n

nn

1 nn

23