17.5 Properties of determinants 491
For interchange of rows
EXAMPLE 17.12Determinant of order 3.
By Example 17.2,
Interchange of rows 1 and 3 gives
= 1 − 51 − 1121 − 141 = 1 − 211 = 1 −D
- Two rows or columns equal
The value of a determinant is zeroif two rows (or two columns)
are equal:
(17.46)
This follows from the antisymmetry property of the determinant. By Property 4, the
sign of the determinant is changed when two rows (or columns) are interchanged.
If the rows (or columns) are identical then such an interchange must leave the
determinant unchanged. Therefore−D 1 = 1 Dand this is possible only ifD 1 = 10.
EXAMPLE 17.13Example of Property 5: rows 1 and 2 are equal.
111
111
234
11
34
11
24
11
23
=− +=−+= 1210
abc
abc
abc
1
11111333= 0
D=−
−
=
−
−
−
−
−
122
013
234
13
34
2
03
24
2
01
23
D=
−
−=
234
013
122
21
DD
2134
12
==×−×=+=−3241 2
D
112
34
==×−×=−1423 2