1. Chapter 3
Determinants
3.1 The Determinant of a Matrix
3.2 Evaluation of a Determinant using
Elementary Operations
3.3 Properties of Determinants
3.4 Application of Determinants
2. 2/62
3.1 The Determinant of a Matrix
the determinant of a 2 × 2 matrix:
Note:
22
21
12
11
a
a
a
a
A
12
21
22
11
|
|
)
det( a
a
a
a
A
A
22
21
12
11
a
a
a
a
22
21
12
11
a
a
a
a
3. 3/62
Ex. 1: (The determinant of a matrix of order 2)
2
1
3
2
2
4
1
2
4
2
3
0
Note: The determinant of a matrix can be positive, zero, or negative.
)
3
(
1
)
2
(
2
3
4
7
)
1
(
4
)
2
(
2
4
4
0
)
3
(
2
)
4
(
0
6
0
6
4. 4/62
Minor of the entry :
The determinant of the matrix determined by deleting the ith row
and jth column of A
Cofactor of :
ij
a
ij
j
i
ij M
C
)
1
(
nn
j
n
j
n
n
n
i
j
i
j
i
i
n
i
j
i
j
i
i
n
j
j
ij
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
M
)
1
(
)
1
(
1
)
1
(
)
1
)(
1
(
)
1
)(
1
(
1
)
1
(
)
1
(
)
1
)(
1
(
)
1
)(
1
(
1
)
1
(
1
)
1
(
1
)
1
(
1
12
11
ij
a
9. 9/62
Thm 3.1: (Expansion by cofactors)
n
j
in
in
i
i
i
i
ij
ij C
a
C
a
C
a
C
a
A
A
a
1
2
2
1
1
|
|
)
det(
)
(
(Cofactor expansion along the i-th row, i=1, 2,…, n )
n
i
nj
nj
j
j
j
j
ij
ij C
a
C
a
C
a
C
a
A
A
b
1
2
2
1
1
|
|
)
det(
)
(
(Cofactor expansion along the j-th row, j=1, 2,…, n )
Let A is a square matrix of order n.
Then the determinant of A is given by
or
10. 10/62
Ex: The determinant of a matrix of order 3
33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
A
33
33
23
23
13
13
32
32
22
22
12
12
31
31
21
21
11
11
33
33
32
32
31
31
23
23
22
22
21
21
13
13
12
12
11
11
)
det(
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
A
11. 11/62
Ex 3: The determinant of a matrix of order 3
14
)
6
)(
1
(
)
8
)(
2
(
)
4
)(
1
(
14
)
3
)(
0
(
)
4
)(
1
(
)
5
)(
2
(
14
)
5
)(
4
(
)
2
)(
3
(
)
1
)(
0
(
14
)
6
)(
1
(
)
3
)(
0
(
)
5
)(
4
(
14
)
8
)(
2
(
)
4
)(
1
(
)
2
)(
3
(
14
)
4
)(
1
(
)
5
)(
2
(
)
1
)(
0
(
)
det(
33
33
23
23
13
13
32
32
22
22
12
12
31
31
21
21
11
11
33
33
32
32
31
31
23
23
22
22
21
21
13
13
12
12
11
11
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
C
a
A
1
0
4
2
1
3
1
2
0
A
6
,
3
,
5
8
,
4
,
2
4
,
5
,
1
33
32
31
23
22
21
13
12
11
2
E
C
C
C
C
C
C
C
C
C
x
Sol:
12. 12/62
Ex 5: (The determinant of a matrix of order 3)
?
)
det(
A
0 2 1
3 1 2
4 0 1
A
1
1
0
2
1
)
1
( 1
1
11
C
Sol:
5
)
5
)(
1
(
1
4
2
3
)
1
( 2
1
12
C
4
0
4
1
3
)
1
( 3
1
13
C
14
)
4
)(
1
(
)
5
)(
2
(
)
1
)(
0
(
)
det( 13
13
12
12
11
11
C
a
C
a
C
a
A
13. 13/62
Ex 4: (The determinant of a matrix of order 4)
2
0
4
3
3
0
2
0
2
0
1
1
0
3
2
1
A ?
)
det(
A
Notes:
The row (or column) containing the most zeros is the best choice
for expansion by cofactors .
15. 15/62
The determinant of a matrix of order 3:
33
32
31
23
22
21
13
12
11
a
a
a
a
a
a
a
a
a
A
32
31
33
32
31
22
21
23
22
21
12
11
13
12
11
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
12
21
33
11
23
32
13
22
31
32
21
13
31
23
12
33
22
11
|
|
)
det(
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
A
A
Add these three products.
Subtract these three products.
17. 17/62
Upper triangular matrix:
Lower triangular matrix:
Diagonal matrix:
All the entries below the main diagonal are zeros.
All the entries above the main diagonal are zeros.
All the entries above and below the main diagonal are zeros.
Note:
A matrix that is both upper and lower triangular is called diagonal.
19. 19/62
Thm 3.2: (Determinant of a Triangular Matrix)
If A is an n × n triangular matrix (upper triangular,
lower triangular, or diagonal), then its determinant is the
product of the entries on the main diagonal. That is
nn
a
a
a
a
A
A
33
22
11
|
|
)
det(
22. 22/62
3.2 Evaluation of a determinant using elementary operations
Thm 3.3: (Elementary row operations and determinants)
)
(
)
( A
r
B
a ij
Let A and B be square matrices.
)
det(
)
det( A
B
)
(
)
( )
(
A
r
B
b k
i
)
det(
)
det( A
k
B
)
(
)
( )
(
A
r
B
c k
ij
)
det(
)
det( A
B
)
)
(
(i.e. A
A
rij
)
)
(
(i.e. )
(
A
k
A
r k
i
)
)
(
(i.e. )
(
A
A
r k
ij
27. 27/62
Notes:
A
E
EA
ij
R
E
)
1
( 1
ij
R
E
A
E
A
R
A
A
r
EA ij
ij
)
(
)
2
( k
i
R
E k
R
E k
i
)
(
A
E
A
R
A
k
A
r
EA k
i
k
i
)
(
)
(
)
(
)
3
( k
ij
R
E 1
)
(
k
ij
R
E
A
E
A
R
A
A
r
EA k
ij
k
ij
)
(
)
(
1
28. 28/62
Determinants and elementary column operations
)
(
)
( A
c
B
a ij
Let A and B be square matrices.
)
det(
)
det( A
B
)
(
)
( )
(
A
c
B
b k
i
)
det(
)
det( A
k
B
)
(
)
( )
(
A
c
B
c k
ij
)
det(
)
det( A
B
)
)
(
(i.e. A
A
cij
)
)
(
(i.e. )
(
A
k
A
c k
i
)
)
(
(i.e. )
(
A
A
c k
ij
Thm: (Elementary column operations and determinants)
30. 30/62
Thm 3.4: (Conditions that yield a zero determinant)
(a) An entire row (or an entire column) consists of zeros.
(b) Two rows (or two columns) are equal.
(c) One row (or column) is a multiple of another row (or column).
If A is a square matrix and any of the following conditions is true,
then det (A) = 0.
36. 36/62
3.3 Properties of Determinants
Notes:
)
det(
)
det(
)
det( B
A
B
A
Thm 3.5: (Determinant of a matrix product)
(1) det (EA) = det (E) det (A)
(2)
(3)
33
32
31
23
22
21
13
12
11
33
32
31
23
22
21
13
12
11
a
a
a
b
b
b
a
a
a
a
a
a
a
a
a
a
a
a
33
32
31
23
23
22
22
22
22
13
12
11
a
a
a
b
a
b
a
b
a
a
a
a
det (AB) = det (A) det (B)
40. 40/62
Ex 3: (Classifying square matrices as singular or nonsingular)
0
A
1
2
3
1
2
3
1
2
0
A
1
2
3
1
2
3
1
2
0
B
A has no inverse (it is singular).
0
12
B B has an inverse (it is nonsingular).
Sol:
Thm 3.7: (Determinant of an invertible matrix)
A square matrix A is invertible (nonsingular) if and only if
det (A) 0
41. 41/62
Ex 4:
?
1
A
0
1
2
2
1
0
3
0
1
A
?
T
A
(a) (b)
4
0
1
2
2
1
0
3
0
1
|
|
A
4
1
1
1
A
A
4
A
AT
Sol:
Thm 3.8: (Determinant of an inverse matrix)
.
)
A
det(
1
)
A
det(
then
invertible
is
If 1
,
A
Thm 3.9: (Determinant of a transpose)
).
det(
)
det(
then
matrix,
square
a
is
If T
A
A
A
42. 42/62
If A is an n × n matrix, then the following statements are equivalent.
(1) A is invertible.
(2) Ax = b has a unique solution for every n × 1 matrix b.
(3) Ax = 0 has only the trivial solution.
(4) A is row-equivalent to In
(5) A can be written as the product of elementary matrices.
(6) det (A) 0
Equivalent conditions for a nonsingular matrix:
43. 43/62
Ex 5: Which of the following system has a unique solution?
(a)
4
2
3
4
2
3
1
2
3
2
1
3
2
1
3
2
x
x
x
x
x
x
x
x
(b)
4
2
3
4
2
3
1
2
3
2
1
3
2
1
3
2
x
x
x
x
x
x
x
x
45. 45/62
3.4 Introduction to Eigenvalues
Eigenvalue problem:
If A is an nn matrix, do there exist n1 nonzero matrices x
such that Ax is a scalar multiple of x？
Eigenvalue and eigenvector:
A：an nn matrix
：a scalar
x： a n1 nonzero column matrix
x
Ax
Eigenvalue
Eigenvector
(The fundamental equation for
the eigenvalue problem)
47. 47/62
Question:
Given an nn matrix A, how can you find the eigenvalues and
corresponding eigenvectors?
Characteristic equation of AMnn:
0
)
I
(
)
I
det( 0
1
1
1
c
c
c
A
A n
n
n
If has nonzero solutions iff .
0
)
I
(
x
A
0
)
I
det(
A
0
)
I
(
x
A
x
Ax
Note:
(homogeneous system)
53. 53/62
3.4 Applications of Determinants
Matrix of cofactors of A:
nn
n
n
n
n
ij
C
C
C
C
C
C
C
C
C
C
2
1
2
22
21
1
12
11
ij
j
i
ij M
C
)
1
(
nn
n
n
n
n
T
ij
C
C
C
C
C
C
C
C
C
C
A
adj
2
1
2
22
12
1
21
11
)
(
Adjoint matrix of A:
54. 54/62
Thm 3.10: (The inverse of a matrix given by its adjoint)
bc
ad
A
)
det(
If A is an n × n invertible matrix, then
)
(
)
det(
1
1
A
adj
A
A
d
c
b
a
A
a
c
b
d
A
adj )
(
)
(
det
1
1
A
adj
A
A
Ex:
a
c
b
d
bc
ad
1
55. 55/62
Ex 1 & Ex 2:
1
A
2
0
1
1
2
0
2
3
1
A
(a) Find the adjoint of A.
(b) Use the adjoint of A to find
,
4
2
0
1
2
11
C
Sol:
,
1
2
1
1
0
12
C 2
0
1
2
0
13
C
ij
j
i
ij M
C
)
1
(
,
6
2
0
2
3
21
C ,
0
2
1
2
1
22
C 3
0
1
3
1
23
C
,
7
1
2
2
3
31
C ,
1
1
0
2
1
32
C 2
2
0
3
1
33
C
56. 56/62
cofactor matrix of A
2
1
7
3
0
6
2
1
4
ij
C
adjoint matrix of A
2
3
2
1
0
1
7
6
4
)
(
T
ij
C
A
adj
2
3
2
1
0
1
7
6
4
3
1
inverse matrix of A
)
(
det
1
1
A
adj
A
A
3
det
A
3
2
3
2
3
1
3
1
3
7
3
4
1
0
2
Check: I
AA
1
57. 57/62
Thm 3.11: (Cramer’s Rule)
1
1
2
12
1
11 b
x
a
x
a
x
a n
n
n
n
nn
n
n
n
n
b
x
a
x
a
x
a
b
x
a
x
a
x
a
2
2
1
1
2
2
2
22
1
21
b
x
A
)
(
)
2
(
)
1
(
,
,
, n
n
n
ij A
A
A
a
A
n
x
x
x
2
1
x
n
b
b
b
2
1
b
0
)
det(
2
1
2
22
21
1
12
11
nn
n
n
n
n
a
a
a
a
a
a
a
a
a
A
(this system has a unique solution)
58. 58/62
)
(
)
1
(
)
1
(
)
2
(
)
1
(
,
,
,
,
,
,
, n
j
j
j A
A
b
A
A
A
A
nn
j
n
n
j
n
n
n
j
j
n
j
j
a
a
b
a
a
a
a
b
a
a
a
a
b
a
a
)
1
(
)
1
(
1
2
)
1
(
2
2
)
1
(
2
21
1
)
1
(
1
1
)
1
(
1
11
nj
n
j
j
j C
b
C
b
C
b
A
2
2
1
1
)
det(
( i.e. )
,
)
det(
)
det(
A
A
x j
j
n
j ,
,
2
,
1
59. 59/62
Pf:
0
)
det(
A
A x = b,
b
x 1
A b
)
(
)
det(
1
A
adj
A
n
nn
n
n
n
n
b
b
b
C
C
C
C
C
C
C
C
C
A
2
1
2
1
2
22
12
1
21
11
)
det(
1
nn
n
n
n
n
n
n
n
C
b
C
b
C
b
C
b
C
b
C
b
C
b
C
b
C
b
A
2
2
1
1
2
22
2
12
1
1
21
2
11
1
)
det(
1
