Unit 18 Determinants
Associated with each square matrix is a number called its determinant.
We defne the determinant and look at some oF its properties in this section.
Defnition 19.1.
G
ivenan(
n
×
n
) matrix
A
we defne the submatrix
A
ij
oF
A
as the matrix obtained by deleting the
i
th
row and
j
th
column oF
A
.
Example 1.
IF
A
=
1213
2131
2534
3789
then
A
11
=
131
534
789
and
A
23
=
123
254
379
Defnition 19.2.
The Determinant DefnitionPart 1
Let
A
=(
a
ij
)bean(
n
×
n
) matrix then the
determinant
oF
A
is defned as :
(i)
det
(
A
)=
a
11
iF
n
= 1. (i.e. iF
A
=[
a
]then
det
([
a
]) =
a
)
(ii)
det
(
A
a
11
a
22

a
12
a
21
iF
n
= 2. (i.e.
det
±
ab
cd
²
=
ad

bc
)
(iii)
det
(
A
a
11
(
detA
11
)

a
12
det
(
A
12
)
...
+(

1)
1+
j
a
1
j
det
(
A
1
j
)+
...+(

1)
1+
n
a
1
n
det
(
A
1
n
)
1
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n
∑
j
=1
(

1)
j
+1
a
1
j
(
det
(
A
1
j
)) if
n>
2
This process is called
expanding the determinant along the frst row
of
A
.
Example 2.
Suppose
A
=
±
12
25
²
Then by (ii) above we have:
det(
A
) = (1)(5)

(2)(2) = 1
Example 3.
Suppose
A
=
123
213
321
Find
det
(
A
)
Solution:
Looking at the de±nition of the determinant we apply (iii) to get:
det
(
A
)=
(

1)
1+1
(1)
det
(
A
11
)+(

1)
1+2
(2)
det
(
A
12

1)
1+3
(3)
det
(
A
13
)
=(1)
det
(
A
11
)

(2)
det
(
A
12
)+(3)
det
(
A
13
)
det
±
13
21
²

(2)
det
±
23
31
²
+(3)
det
±
32
²
=(1)(

5)

(2)(

4) + (3)(1) = 6
Suppose we are asked to ±nd the determinant of a matrix such as
A
=
1234
2131
2000
3789
Then
det
(
A
)=(

1)
2
(1)
det
(
A
11

1)
3
(2)
det
(
A
12

1)
4
(3)
det
(
A
13

1)
5
(4)
det
(
A
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 Spring '10
 lit
 Calculus, Determinant, Diagonal matrix, Triangular matrix, Det

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