Section 10.4 Determinants
Determinants are important mathematical tools that can frequently be used to analyze and solve
linear systems. There are various methods for finding determinants. Some of the simpler methods
only apply to 2
x
2 and 3
x
3 matrices.
Determinants of
2
x
2
matrices
If
D
=
a
b
c
d
is a 2 by 2 matrix, then the
determinant
of
D
is
•
det
(
D
) =
a
b
c
d
=
ad

bc
Example.
Find the determinant of
D
=
1

1
2

3
Solution.
1

1
2

3
= (1)(

3)

(

1)(2) =

1
Determinants of
3
x
3
matrices
If
D
=
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
is a 3
x
3 matrix, then the determinant of
D
is
•
det
(
D
) =
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
=
a
11
a
22
a
33
+
a
12
a
23
a
31
+
a
13
a
21
a
32

(
a
13
a
22
a
31
+
a
12
a
21
a
33
+
a
11
a
23
a
32
)
The following is a good device for remembering how to evaluate 3
x
3 determinants.
Example.
Find the determinant of
D
=
1

1
0
2
1
1

1

2

1
Solution.
1

1
0
2
1
1

1

2

1
= (1)(1)(

1) + (

1)(1)(

1) + (0)(2)(

2)

((0)(1)(

1) + (

1)(2)(

1) + (1)(1)(

2))
1
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=

1 + 1 + 0

(0 + 2

2) = 0
Important note. The methods shown above for 2
x
2 and 3
x
3 determinants does NOT apply to 4
x
4
or higherorder determinants.
Cramer’s Rule
Cramer’s rule is a method that can be used for solving linear systems when the number of unknowns
is the same as the number of equations and the determinant of the system’s coefficient matrix is not
equal to zero.
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 Fall '11
 Kutter
 Algebra, Determinant, Linear Systems, Matrices, Howard Staunton, Cramer, A21, a32, a13

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