The Chemistry Maths Book, Second Edition

17.5 Properties of determinants 489

(17.40)

2. Multiplication by a constant

If all the elements of any row (or column) are multiplied by the

same factor λ, the value of the new determinant is λtimes the

value of the old determinant:

(17.41)

EXAMPLE 17.10Examples of Property 2.

A special case isλ 1 = 10 : if all the elements of a row (or column) are zero, the value of

the determinant is zero:

(17.42)

3. Addition rule

If all the elements of any row (or column) are written as the sum

of two terms the determinant can be written as the sum of two

determinants:

(17.43)

ad b c

ad bc

ad b c

abc

abc

ab

11

11

22

22

33

33

1

11

222

3

• 
  

• 
  

• 
  

=

333

1

11

222

333

c

dbc

dbc

dbc

• 
  

000

0

2 22

332

abc

abc

=

=×× 223 =××× =

111

143 2233

111

143

123

144

369

246 2

8

12

3

9

27

189

31227

2

13

19

327

22

11

14

36

==×

λλλ

λ

abc

abc

abc

abc

abc

abc

1

11

222

333

1

11

222

333

=

abc

abc

abc

aaa

bbb

ccc

1

11

222

333

1

23

123

123

=