492 Chapter 17Determinants
- Proportional rows or columns
The value of a determinant is zeroif one row (or column) is a
multiple of another row (or column):
(17.47)
This follows from Property 2 for the multiple of a determinant, and Property 5 for two
equal rows.
EXAMPLE 17.14Example of Property 6: column 1 is a multiple of column 2.
- Addition of rows or columns
The value of a determinant is unchangedwhen a multiple of any
row (or column) is added to any other row (or column):
(17.48)
This follows from Property 3 for the addition of determinants and Property 6 for
one column equal to a multiple of an other.
EXAMPLE 17.15Example of Property 7. Three times column 2 has been added to
column 1.
411
834
15 5 6
131 1 1
233 3 4
035 5 6
111
= 2
+×
+×
+×
=
()
()
()
334
056
3
111
334
556
111
234
056
+=
=
abc
abc
abc
1
11222333abbc
a bbc
abbc
abc
abc
11
11222233331
11222=
λ
λ
λ aabc
bbc
bbc
bbc
3331
11222333λ
λ
λ
423
214
635
2
223
114
335
== 0
λλλ
λ
abc
abc
abc
abc
abc
abc
2
222223332
22222333== 0