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Unit 19 Properties of Determinants
Theorem 20.1.
Suppose
A
1
and
A
2
are identical
n
×
n
matrices with the
exception that one row ( or column ) of
A
2
is obtained by multiplying the
corresponding row ( or column ) of
A
1
by some nonzero constant
c
.Th
en
det
(
A
2
)=
c
(
det
(
A
1
))
This theorem may be used in a variety of situations to make a seemingly
diﬃcult determinant ( due to large numbers or fractions involved ) into a
much easier one, as the following examples demonstrate.
Example 1.
It is easily seen that
det
293
40 0
31
−
1
=48
Therefore
det
693
12 0
0
91
−
1
=3(
−
48) =
−
144
Example 2.
Find
detA
where
A
=
2 04
1
−
62
0
39
1
2
Solution:
det
1
−
1
2
=2
×
det
10
2
1
−
1
2
=3
×
2
×
det
2
1
−
13
4
×
3
×
2
×
det
2
1
−
22
11
4
×
3
×
3
×
2
×
det
1
1
−
21
2
(expanding along the ±rst row we get)
=36
{
1[
−
4
−
1]
−
0[2
−
1]+1[1+2]
}
= 36(
−
2)=
−
72
1
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View Full Document Example 3.
Find the determinant of
1
3
0
3
4
2
5
−
1
3
2
1
8

3
4
5
4
Solution:
By taking a factor of
1
12
from row 1, a factor of
1
10
from row 2, and
one of
1
8
from row 3, we see that
det
1
3
0
3
4
2
5
−
1
3
2
1
8

3
4
5
4
=
1
12
×
1
10
×
1
8
×
det
40
9
4
−
10 15
−
16
1
0
=
1
12
×
1
10
×
1
8
×
(
−
2)
×
det
40 9
451
5
−
131
0
=
−
1
480
×
83 =
−
83
480
Remark.
Be warned that it is
not
true in general that if
c
is a constant then
det
(
cA
)=
cdet
(
A
). Since
cA
is formed by multiplying every entry and hence
every row of
A
by
c
we may take out a factor of
c
from each row, so for any
n
×
n
matrix
A
:
det
(
cA
c
n
det
(
A
)
So, for example if
A
is a 3
×
3 matrix, then
det
(2
A
)=2
3
det
(
A
)=8
det
(
A
).
Theorem 20.2. More Facts About Determinants
Suppose A and B are square
n
×
n
matrices, then:
1. If A has one row (column) that is a scalar multiple of another row
(column) of A then
det
(
A
)=0
.
2. If two rows ( columns ) of A are interchanged then the determinant
changes by a factor of 1. (i.e. the sign is reversed).
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This note was uploaded on 04/15/2010 for the course MATH 20C taught by Professor Lit during the Spring '10 term at UCLA.
 Spring '10
 lit
 Calculus, Determinant, Matrices

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