The Chemistry Maths Book, Second Edition

17.6 Reduction to triangular form 493

8. Linearly-dependent rows or columns

The value of a determinant is zeroif the rows (or columns)

are linearly dependent; that is, if a row (or column) is a linear

combination of the others.

(17.49)

This follows from Properties 5, 6 and 7. It is the general condition for the value of a

determinant to be zero.

0 Exercises 22–24

9. Derivative of a determinant

If the elements of a determinant Dare differentiable functions,

the derivative D′of Dcan be written

D′ 1 = 1 D

1

1 + 1 D

2

1 + 1 D

3

1 +1-1+ 1 D

n

(17.50)

where D

i

is obtained from Dby differentiation of the elements of

the ith row.

EXAMPLE 17.16Differentiation of a third-order determinant.

If the elements are functions of x,

0 Exercise 25

17.6 Reduction to triangular form

It is demonstrated in Example 17.7 that a ‘triangular’ determinant has value equal to

the product of its diagonal elements,

d

dx

aaa

bbb

ccc

da

dx

da

dx

da

dx

bbb

1

23

123

123

123

1 23

=

cccc

aaa

db

dx

db

dx

db

dx

ccc

aaa

b

123

1

23

1

23

1

23

1

23

++

1123

1

23

bb

dc

dx

dc

dx

dc

dx

λμ

λμ

λμ

λ

bcbc

bcbc

bcbc

bbc

b

11

11

22

22

33

33

1

11

2

• 
  

• 
  

• 
  

= bbc

bbc

cbc

cbc

cbc

22

333

1

11

2

22

3

33

+=μ 0