CHAPTER 3
DETERMINANTS
3.1 The Determinant of a Matrix
3.2 Determinant and Elementary Operations
3.3 Properties of Determinants
3.4 Application of Determinants
Elementary Linear Algebra
R. Larson (7 Edition)
投影片設計編製者
淡江大學 電機系 翁慶昌 教授
CH 3 Linear Algebra Applied
Volume of a Solid (p.108)
Engineering and Control (p.124)
Sudoku (p.114)
Planetary Orbits (p.135)
Textbook Publishing (p.137)
2/64
3.1 The Determinant of a Matrix

the determinant of a 2 × 2 matrix:
 a 11
A 
 a 21
 det( A )

a 12 

a 22 

|A|

a 11 a 22

a 21 a 12
Note:
 a 11

 a 21
a 12 
a 11
  a
a 22 
21
a 12
a 22
Elementary Linear Algebra: Section 3.1, p.104
3/64


Ex. 1: (The determinant of a matrix of order 2)
2
3
1
2
2
1
4
2
0
3
2
4
 2 ( 2 )  1(  3 )  4  3
7
 2 ( 2 )  4 (1)  4  4  0
 0 ( 4 )  2 ( 3)  0  6   6
Note: The determinant of a matrix can be positive, zero, or negative.
Elementary Linear Algebra: Section 3.1, p.104
4/64

Minor of the entry aij :
The determinant of the matrix determined by deleting the ith row
and jth column of A
a 11
a 12


M ij 
a 1( j 1 )



a1n
a ( i 1 )1

a ( i 1 )( j  1 )
a ( i  1 )( j  1 )

a ( i 1 ) n
a ( i 1 )1

a ( i  1 )( j 1 )
a ( i  1 )( j  1 )

a ( i 1 ) n


a n ( j 1 )
a n ( j 1 )


a n1

a 1 ( j 1 )


a nn
Cofactor of aij :
C ij  (  1)
i j
M ij
Elementary Linear Algebra: Section 3.1, p.105
5/64

Ex:
 a 11
A   a 21

 a 31
 M
21

a 32
a 13 
a 23 

a 33 
a 12
a 13
a 32
a 33
a 12
a 22
 C 21  (  1)
2 1
M 21   M 21
Elementary Linear Algebra: Section 3.1, p.105
M 22 
a 11
a 13
a 31
a 33
C 22  (  1)
22
M
22
 M
22
6/64

Notes: Sign pattern for cofactors




 







 
3 × 3 matrix























4 × 4 matrix








 

































n ×n matrix
Notes:
Odd positions (where i+j is odd) have negative signs, and
even positions (where i+j is even) have positive signs.
Elementary Linear Algebra: Section 3.1, p.105
7/64

Ex 2: Find all the minors and cofactors of A.
0

A 3

 4
2
1
0
1

2

1 
Sol: (1) All the minors of A.
 M 11 
M 21 
M 31 
1
2
0
1
2
1
0
  1, M 12 
 2,
1
2
1
1
2
 5,
Elementary Linear Algebra: Section 3.1, p.105
M 22 
M 32
3
4
0
2
  5, M
1
1
4
1
0
1
3
2
  4, M
13
23


  3 , M 33 
3
1
4
0
0
2
4
1
4
 8
0
2
3
1
 6
8/64
Sol: (2) All the cofactors of A.

C ij  (  1)
 C 11  
C 21  
C 31  
i j
M
1
2
0
1
2
0
1
ij
22
1
1
1
2
2
4
1
0
1
4
1
0
1
3
2
  1, C 12  
  2, C  
2
3
 5 , C 32  
Elementary Linear Algebra: Section 3.1, p.105
 5 , C 13  
  4, C  
23
 3,
C 33  
3
1
4
0
0
2
4
1
0
2
3
1
4
8
 6
9/64

Thm 3.1: (Expansion by cofactors)
Let A is a square matrix of order n.
Then the determinant of A is given by
n
(a )
det( A )  | A |
a
ij
C ij  a i1C i1  a i 2 C i 2    a in C in
j 1
(Cofactor expansion along the i-th row, i=1, 2,…, n )
or
n
(b )
det( A )  | A |
a
ij
C ij  a1 j C 1 j  a 2 j C 2 j    a nj C nj
i 1
(Cofactor expansion along the j-th row, j=1, 2,…, n )
Elementary Linear Algebra: Section 3.1, p.107
10/64

Ex: The determinant of a matrix of order 3
 a 11
A   a 21

 a 31
a 12
a 22
a 32
a 13 
a 23 

a 33 
 det( A )  a 11 C 11  a 12 C 12  a 13 C 13
 a 21 C 21  a 22 C 22  a 23 C 23
 a 31 C 31  a 32 C 32  a 33 C 33
 a 11 C 11  a 21 C 21  a 31 C 31
 a 12 C 12  a 22 C 22  a 32 C 32
 a 13 C 13  a 23 C 23  a 33 C 33
Elementary Linear Algebra: Section 3.1, Addition
11/64

Ex 3: The determinant of a matrix of order 3
0

A 3

 4
2
1
0
1

2

1 
Ex 2
 C 1 1   1, C 1 2  5 ,
C 13  4
C 21   2 , C 22   4 , C 23  8
C 31  5 ,
C 32  3, C 33   6
Sol:
 det( A )  a 1 1 C 1 1  a 1 2 C 1 2  a 1 3 C 1 3  ( 0 )(  1)  ( 2 )( 5 )  (1)( 4 )  14
 a 2 1 C 2 1  a 2 2 C 2 2  a 2 3 C 2 3  ( 3 )(  2 )  (  1)(  4 )  ( 2 )( 8 )  14
 a 3 1 C 3 1  a 3 2 C 3 2  a 3 3 C 3 3  ( 4 )( 5 )  ( 0 )( 3 )  (1)(  6 )  14
 a 1 1 C 1 1  a 2 1 C 2 1  a 3 1 C 3 1  ( 0 )(  1)  ( 3 )(  2 )  ( 4 )( 5 )  14
 a 1 2 C 1 2  a 2 2 C 2 2  a 3 2 C 3 2  ( 2 )( 5 )  (  1)(  4 )  ( 0 )( 3 )  14
 a 1 3 C 1 3  a 2 3 C 2 3  a 3 3 C 3 3  (1)( 4 )  ( 2 )( 8 )  (1)(  6 )  14
Elementary Linear Algebra: Section 3.1, p.106
12/64

Ex 5: (The determinant of a matrix of order 3)
0

A 3

 4
1

2

1 
2
1
4
 det( A )  ?
Sol:
C 11  (  1)
1
2
4
1
3
1
4
4
1 1
C 13  (  1)
1 3
7
C 12  (  1)
1 2
3
2
4
1
 (  1)(  5 )  5
 8
 det( A )  a 11 C 11  a 12 C 12  a 13 C 13
 ( 0 )( 7 )  ( 2 )( 5 )  (1)(  8 )
2
Elementary Linear Algebra: Section 3.1, p.108
13/64


Notes:
The row (or column) containing the most zeros is the best choice
for expansion by cofactors .
Ex 4: (The determinant of a matrix of order 4)
 1

1

A 
 0

 3
2
3
1
0
2
0
4
0
0 

2

3 

 2
Elementary Linear Algebra: Section 3.1, p.107
 det( A )  ?
14/64
Sol:
det( A )  ( 3 )( C 13 )  ( 0 )( C 23 )  ( 0 )( C 33 )  ( 0 )( C 43 )
 3C 13
 3 (  1)
1 3
1
1
2
0
2
3
3
4
2
1

2 1
 3  ( 0 )(  1)
4

2
2
 ( 2 )(  1)
22
1
2
3
2
 ( 3 )(  1)
23
1
3
1

4
 30  ( 2 )( 1)(  4 )  ( 3 )(  1)(  7 ) 
 ( 3 )( 13 )
 39
Elementary Linear Algebra: Section 3.1, p.107
15/64

The determinant of a matrix of order 3:
Subtract these three products.
 a 11

A  a 21

 a 31
a 12
a 22
a 32
a 13 

a 23

a 33 
a 11
a 12
a 13
a 11
a 12
a 21
a 22
a 23
a 21
a 22
a 31
a 32
a 33
a 31
a 32
Add these three products.
 det( A )  | A | a 11 a 22 a 33  a 12 a 23 a 31  a 13 a 21 a 32  a 31 a 22 a 13
 a 32 a 23 a 11  a 33 a 21 a 12
Elementary Linear Algebra: Section 3.1, p.108
16/64

Ex 5:
–4 0
0
A  3

 4
2
1
4
1
2

1 
6
0
2
3
1
4
4
0 16 –12
 det( A )  | A | 0  16  12  (  4 )  0  6  2
Elementary Linear Algebra: Section 3.1, p.108
17/64

Upper triangular matrix:
All the entries below the main diagonal are zeros.

Lower triangular matrix:
All the entries above the main diagonal are zeros.

Diagonal matrix:
All the entries above and below the main diagonal are zeros.

Note:
A matrix that is both upper and lower triangular is called diagonal.
Elementary Linear Algebra: Section 3.1, p.109
18/64

Ex:
 a 11 a 12 a 13 
 0 a 22 a 23 
 0

0
a
33


0 
 a 11 0
 a 21 a 22 0 
a

 31 a 32 a 33 
0 
 a 11 0
 0 a 22 0 
 0
0 a 33 

upper triangular
lower triangular
diagonal
Elementary Linear Algebra: Section 3.1, p.109
19/64

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
det( A )  | A | a 11 a 22 a 33  a nn
Elementary Linear Algebra: Section 3.1, p.109
20/64

Ex 6: Find the determinants of the following triangular matrices.
(a)
 2
 4
A 
 5
 1

0
0
2
0
6
1
5
3
0
0

0
3 
Sol:
 1

0

(b) B   0

 0
 0

0
0
0
3
0
0
0
2
0
0
0
4
0
0
0
0

0

0

0
 2 
(a) |A| = (2)(–2)(1)(3) = –12
(b) |B| = (–1)(3)(2)(4)(–2) = 48
Elementary Linear Algebra: Section 3.1, p.109
21/64
Key Learning in Section 3.1

Determine whether two matrices are equal.

Add and subtract matrices and multiply a matrix by a scalar.

Multiply two matrices.

Use matrices to solve a system of linear equations.

Partition a matrix and write a linear combination of column
vectors..
22/64
Keywords in Section 3.1

determinant : 行列式

minor : 子行列式

cofactor : 餘因子

expansion by cofactors : 餘因子展開

upper triangular matrix: 上三角矩陣

lower triangular matrix: 下三角矩陣

diagonal matrix: 對角矩陣
23/64
3.2 Evaluation of a determinant using elementary operations

Thm 3.3: (Elementary row operations and determinants)
Let A and B be square matrices.
(a )
B  rij ( A )

det( B )   det( A )
(i.e. rij ( A )   A )
(b)
B  ri
( A)

det( B )  k det( A )
(i.e. ri
(c)
B  rij ( A )

det( B )  det( A )
(i.e. rij ( A )  A )
(k )
(k )
Elementary Linear Algebra: Section 3.2, p.113
(k )
( A)  k A )
(k )
24/64

Ex:
1
A  0

 1
2
1
2
4

A1  0

 1
8
A1  r1
( A)
(4)
1
2
A 2  r12 ( A )
( 2 )
A 3  r12
( A)
3
4

1 
det( A )   2
12 

4

1 
0

A2  1

 1
1
2
2
4

3

1 
 det( A1)  det( r1
(4)
 1

A3   2

 1
2
3
2
3 

2

1 
( A ))  4 det( A )  ( 4 )(  2 )   8
 det( A 2 )  det( r12 ( A ))   det( A )   (  2 )  2
( 2 )
 det( A 3 )  det( r12
Elementary Linear Algebra: Section 3.2, Addition
( A ))  det( A )   2
25/64

Notes:
det( rij ( A ))   det( A )
det( ri
(k )
 det( A )   det( rij ( A ))
( A ))  k det( A )  det( A ) 
1
k
det( rij ( A ))  det( A )
(k )

Elementary Linear Algebra: Section 3.2, Addition
det( ri
(k )
( A ))
det( A )  det( rij ( A ))
(k )
26/64
Note:
A row-echelon form of a square matrix is always upper triangular.

Ex 2: (Evaluation a determinant using elementary row operations)
 2  3 10 
A  1 2  2 
 0 1  3 
 det( A )  ?
Sol:
2
3
10
1
r1 2
det( A )  1
2
2   2
0
1
3
Elementary Linear Algebra: Section 3.2, p.113
0
2
2
3
10
1
3
27/64
( 2 )
r1 2
1
2
  0
0
2
7
14
1
3
(1 )
r2 7
 (  1)(
1
1
7
1
2
2
) 0
1
2
0
1
3
1
2
2
1
 ( 7 )( ) 0
1
 2  ( 7 )(  3 )(1)(1)(1)   21
0
0
( 1 )
3
r3
1
3
1
Elementary Linear Algebra: Section 3.2, p.113
28/64

Notes:
EA  E A
(1)
E  R ij
 E  R ij   1
 EA  rij  A    A  R ij A  E A
(2)
E  Ri
(k )
 E  Ri
(k )
 EA  ri
(3)
E  R ij
(k )
(k )
 E  R ij
(k )
 EA  rij
(k )
Elementary Linear Algebra: Section 3.2, Addition
k
A
 k A  Ri
(k )
A  E A
1
A
 1 A  R ij
(k )
A  E A
29/64

Determinants and elementary column operations

Thm: (Elementary column operations and determinants)
Let A and B be square matrices.
(a )
B  c ij ( A )

det( B )   det( A )
(i.e. c ij ( A )   A )
(b )
B  ci ( A)

det( B )  k det( A )
(i.e. c i ( A )  k A )
(c )
B  c ij ( A )

det( B )  det( A )
(i.e. c ij ( A )  A )
(k )
(k )
Elementary Linear Algebra: Section 3.2, p.114
(k )
(k )
30/64

Ex:
2

A 4

 0
1
1

A1  2

 0
1
(
0
0
1
0
0
)
A 1  c1 ( A )
2
A 2  c 12 ( A )
A 3  c 23 ( A )
(3)
 3

1

2 
det( A )   8
 3

1

2 
1

A2  0

 0
2
4
0
 3

1

2 
 det( A 1 )  det( c 1 ( A )) 
(4)
2

A3  4

 0
1
1
0
0
0

1

2 
1
det( A )  ( )(  8 )   4
2
2
 det( A 2 )  det( c 12 ( A ))   det( A )   (  8 )  8
 det( A 3 )  det( c 23 ( A ))  det( A )   8
Elementary Linear Algebra: Section 3.2, p.114
(3)
31/64

Thm 3.4: (Conditions that yield a zero determinant)
If A is a square matrix and any of the following conditions is true,
then det (A) = 0.
(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).
Elementary Linear Algebra: Section 3.2, p.115
32/64

Ex:
1
2
3
1
4
0
1
1
1
0
0
0 0
2
5
0 0
2
2
2 0
4
5
6
3
6
0
4
5
6
1
4
2
1
2
3
1
8
1
5
2 0
4
5
6 0
2
10
5 0
1
6
2
2
4
3
12
6
Elementary Linear Algebra: Section 3.2, Addition
6
4
33/64

Note:
Cofactor Expansion
Row Reduction
Order n
Additions
Multiplications
Additions
Multiplications
3
5
9
5
10
5
119
205
30
45
10
3,628,799
6,235,300
285
339
Elementary Linear Algebra: Section 3.2, p.116
34/64

Ex 5: (Evaluating a determinant)
 3
A  2

  3
Sol:
2 
 1

6 
5
4
0
3
det( A )  2
3
5
4
det( A )  2
3
6
3
5
4
2
4
3
3
0
0
5
4
4
3
3 1
5
4
2
1 2
(
4
)
r 5
12
1 
0
 ( 5 )(  1)
(2)
C13
1 
0
 (  3 )(  1)
3
2
6
2
5
3
5
3
6
 (  3 )(  1)  3
3
5
2
2
5
0
3
5
3
0
6
 (  5 )( 
Elementary Linear Algebra: Section 3.2, p.116
3
)3
5
35/64

Ex 6: (Evaluating a determinant)


A 


2 0
1 3  2
2
1 3 2  1
1 0 1 2
3
3  1 2 4  3
1 1 3 2
0 
Sol:
2 0
1 3 2
2 0
1 3 2
2
1 3 2 1
 2 1 3 2 1
det( A ) 
1 0 1 2
3 (1 )
1 0 1 2
3
r2 4
3  1 2 4  3 r ( 1) 1 0 5 6  4
25
1 1 3 2
0
3 0 0 0
1
 (1)(  1)
22
2 1
1 1
1 5
3 0
3 2
2 3
6 4
0 1
Elementary Linear Algebra: Section 3.2, p.117
36/64
C
( 3 )
41

8
1
3
2
8
1
2
3
13
5
6
4
0
0
0
1
 5(  1)
1 3
8
1
13
5
 (1)(  1)
4 4
8
1
3
0
8
1
2 ( 1 )  8
0
5
1
2
5
6
r
21
13
5
6
13
 ( 5 )(  27 )
  135
Elementary Linear Algebra: Section 3.2, p.117
37/64
Key Learning in Section 3.2

Determine whether two matrices are equal.

Add and subtract matrices and multiply a matrix by a scalar.

Multiply two matrices.

Use matrices to solve a system of linear equations.

Partition a matrix and write a linear combination of column
vectors.
38/64
Keywords in Section 3.2

determinant : 行列式

elementary row operation: 基本列運算

row equivalent: 列等價

elementary matrix: 基本矩陣

elementary column operation: 基本行運算

column equivalent: 行等價
39/64
3.3 Properties of Determinants

Thm 3.5: (Determinant of a matrix product)
det (AB) = det (A) det (B)

Notes:
(1) det (EA) = det (E) det (A)
(2) det( A  B )  det( A )  det( B )
(3)
a 11
a 12
a 22  b 22
a 22  b 22
a 31
a 32
a 13
a 11
a 12
a 13
a 11
a 12
a 13
a 23  b 23  a 21
a 22
a 23  b 21
b 22
b 23
a 31
a 32
a 33
a 32
a 33
Elementary Linear Algebra: Section 3.3, p.120
a 33
a 31
40/64

Ex 1: (The determinant of a matrix product)
1

A 0

 1
2
3
0
2

2

1 
2

B  0

 3
0
1
1
1 

2

 2 
Find |A|, |B|, and |AB|
Sol:
1
2
2
| A | 0
3
2  7
1
0
1
Elementary Linear Algebra: Section 3.3, p.120
2
| B | 0
3
0
1
1
1
 2  11
2
41/64
2
1

AB  0

 1
3
0
8
 | AB | 6
5

2 2

2 0

1   3
4
1
1
0
1
1
1  8
 
2  6
 
 2   5
4
1
1
1 

 10

 1 
1
 10   77
1
Check:
|AB| = |A| |B|
Elementary Linear Algebra: Section 3.3, p.120
42/64
Thm 3.6: (Determinant of a scalar multiple of a matrix)

If A is an n × n matrix and c is a scalar, then
det (cA) = cn det (A)

Ex 2:
 10
A   30

  20
 20
0
 30
40 
50  ,

10 
1
2
3
0
2
3
4
5 5
1
Find |A|.
Sol:
 1
A  10  3

  2
2
0
3
1
4
3


A

10
3
5

2
1 
Elementary Linear Algebra: Section 3.3, p.121
2
0
3
4
5  (1000 )( 5 )  5000
1
43/64

Thm 3.7: (Determinant of an invertible matrix)
A square matrix A is invertible (nonsingular) if and only if
det (A)  0

Ex 3: (Classifying square matrices as singular or nonsingular)
0

A 3

 3
2
2
2
 1

1

 1
0

B  3

 3
2
2
2
 1

1

1 
Sol:
A 0

A has no inverse (it is singular).
B   12  0

B has an inverse (it is nonsingular).
Elementary Linear Algebra: Section 3.3, p.122
44/64

Thm 3.8: (Determinant of an inverse matrix)
1
1
If A is invertible , then det( A ) 
.
det( A )

Thm 3.9: (Determinant of a transpose)
If A is a square matrix, then det( A )  det( A ).
T

Ex 4:
1

A 0

 2
0
1
1
3

2

0 
(a) A  1  ?
(b)
Sol:
1

| A | 0
2
0
1
1
3
2 4
0
Elementary Linear Algebra: Section 3.3, pp.122-124

A
1

1
A
A
T

A
T
?
1
4
 A 4
45/64

Equivalent conditions for a nonsingular matrix:
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
Elementary Linear Algebra: Section 3.3, p.123
46/64

Ex 5: Which of the following system has a unique solution?
(a)
2 x2

x3

1
3 x1

2 x2

x3

4
3 x1

2 x2

x3

4
2 x2

x3

1
(b)
3 x1

2 x2

x3

4
3 x1

2 x2

x3

4
Elementary Linear Algebra: Section 3.3, p.123
47/64
Sol:
(a)
Ax  b
A 0


This system does not have a unique solution.
(b) B x  b


B   12  0
This system has a unique solution.
Elementary Linear Algebra: Section 3.3, p.123
48/64
Key Learning in Section 3.3

Determine whether two matrices are equal.

Add and subtract matrices and multiply a matrix by a scalar.

Multiply two matrices.

Use matrices to solve a system of linear equations.

Partition a matrix and write a linear combination of column
vectors.
49/64
Keywords in Section 3.3

determinant: 行列式

matrix multiplication: 矩陣相乘

scalar multiplication: 純量積

invertible matrix: 可逆矩陣

inverse matrix: 反矩陣

nonsingular matrix: 非奇異矩陣

transpose matrix: 轉置矩陣
50/64
3.4 Applications of Determinants

Matrix of cofactors of A:
 C 11

C 21

C ij  
 

 C n1

C 12

C 22


Cn2

C 1n 

C 2n

 

C nn 
C ij  (  1)
i j
M ij
Adjoint matrix of A:
adj ( A )  C ij 
T
 C 11

C 12


 

C 1n
Elementary Linear Algebra: Section 3.4, p.128
C 21

C 22


C 2n

C n1 

Cn2

 

C nn 
51/64

Thm 3.10: (The inverse of a matrix given by its adjoint)
If A is an n × n invertible matrix, then
A
1
1

adj ( A )
det( A )

Ex:
a
A 
c
b

d
 det( A )  ad  bc
 d
adj ( A )  
 c
 b

a 
Elementary Linear Algebra: Section 3.4, p.129
 A
1

1
det  A 
adj ( A )
 d

ad  bc   c
1
 b
a 
52/64

Ex 1 & Ex 2:
 1

A  0

 1
Sol:

3
2
0
2 

1

 2 
C ij  (  1)
 C 11  
C 21  
C 31  
i j
(a) Find the adjoint of A.
(b) Use the adjoint of A to find A 1
M
2
1
0
2
3
2
0
2
ij
 4 , C 12  
 6,
3
2
2
1
 7,
C 22  
C 32  
Elementary Linear Algebra: Section 3.4, pp.128-129
0
1
1
2
1
2
1
2
1
2
0
1
 1, C 13  
 0 , C 23  
 1, C 33  
0
2
1
0
1
3
1
0
1
3
0
2
2
3
2
53/64
 cofactor matrix of A
C 
ij

4

 6

 7
1
0
1
adjoint matrix of A
2

3

2 
adj ( A )  C ij 
T
4
 1

 2
6
0
3
7
1

2 
inverse matrix of A
A
1

1
det  A 
4
 13  1

 2


Check:
AA
6
0
3
1

adj ( A )
7   43
1    13
 
2   23
2
0
1
det  A   3

1
3

2

3
7
3
I
Elementary Linear Algebra: Section 3.4, pp.128-129
54/64

Thm 3.11: (Cramer’s Rule)
a 11 x 1  a 12 x 2    a 1 n x n  b1
a 21 x 1  a 22 x 2    a 2 n x n  b 2

a n 1 x 1  a n 2 x 2    a nn x n  b n
A  a ij 
Ax  b
det( A ) 
nn
 A
(1 )
a 11
a 12

a1 n
a 21
a 22

a2n


a n1
an2


Elementary Linear Algebra: Section 3.4, p.131
,A
(2)
, , A
(n)

 b1 
 x1 
b 
x 
2
2
x   b  
 
  
 
 
x
 bn 
 n
0
(this system has a unique solution)
a nn
55/64

Aj  A
(1 )
 a 11
a
21
 
 
a
 n1
,A
(2)
, , A
( j 1 )
, b, A
( j 1)
, , A
(n)

a 1 ( j 1 )
b1
a 1( j 1 )


a 2 ( j 1 )
b2
a 2 ( j 1 )

a n ( j 1 )



a n ( j 1 )
bn

a1n 
a2n 

 
a nn 
( i.e. det( A j )  b1C 1 j  b2 C 2 j    bn C n j )
 xj 
det( A j )
det( A )
,
j  1 , 2 , , n
Elementary Linear Algebra: Section 3.4, p.131
56/64

Pf:
det( A )  0
A x = b,
1
 x A b
1

adj ( A ) b
det( A )
 C 11

C 12
1


det( A )  

C 1n
 b1C 11
b C
1
 1 12

det( A ) 

 b1C 1 n
C 21

C 22


C 2n

C n 1   b1 
 
C n 2 b2
 
   
 
C nn   b n 

b 2 C 21




b 2 C 22







Elementary Linear Algebra: Section 3.4, p.131

b2 C 2 n

bn C n1 
bn C n 2 



b n C nn 
57/64
 xj 

1
det( A )
( b1C 1 j  b 2 C 2 j    b n C nj )
det( A j )
det( A )
j  1 ,2 , , n
Elementary Linear Algebra: Section 3.4, p.131
58/64

Ex 4: Use Cramer’s rule to solve the system of linear equations.
x

2y
2x
3x
Sol:

4y
1

3z

1

z

0

4z

2
2
det( A )  2
0
3
3
1  10
4
4
1
3
1
1   15 ,
3
2
4
x
det( A )

5
y 
det( A1 )  0
0
1
0
4
2
2
det( A2 )  2
det( A1 )
1
det( A2 )
det( A )
Elementary Linear Algebra: Section 3.4, p.131
det( A3 )  2

3
2
3
1 8
4
4
2
1
0
0   16
3
4
z
det( A3 )
det( A )
2

8
5
59/64
Keywords in Section 3.4

matrix of cofactors : 餘因子矩陣

adjoint matrix : 伴隨矩陣

Cramer’s rule : Cramer 法則
60/64
3.1 Linear Algebra Applied

Volume of a Solid
If x, y and z are continuous functions of u, v, and w
with continuous first partial derivatives, then the
Jacobians J(u, v) and J(u, v, w) are defined as the
determinants
J (u , v ) 
x
x
u
v
y
y
u
v
and
J (u , v, w ) 
x
x
x
u
v
w
y
y
y
u
v
w
z
z
z
u
v
w
.
And one practical use of Jacobians is in finding the
volume of a solid region. In Section 3.4, you will
study a formula, which also uses determinants, for
finding the volume of a tetrahedron. In the Chapter 3
Review, you are asked to find the Jacobian of a given
set of functions. (See Review Exercises 49–52.)
Elementary Linear Algebra: Section 3.1, p.108
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3.2 Linear Algebra Applied

Sudoku
In the number-placement puzzle Sudoku, the object is
to fill out a partially completed 99 grid of boxes
with numbers from 1 to 9 so that each column, row,
and 33 sub-grid contains each of these numbers
without repetition. For a completed Sudoku grid to be
valid, no two rows (or columns) will have the numbers
in the same order. If this should happen in a row or
column, then the determinant of the matrix formed by
the numbers in the grid will be zero. This is a direct
result of condition 2 of Theorem 3.4.
Elementary Linear Algebra: Section 3.2, p.114
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3.3 Linear Algebra Applied

Engineering and Control
Systems of linear differential equations often arise in
engineering and control theory. For a function f(t) that
is defined for all positive values of t the Laplace
transform of f(t) is given by
F (s) 


e
 st
f (t ) d t .
0
Laplace transforms and Cramer’s Rule, which uses
determinants to solve a system of linear equations, can
often be used to solve a system of differential
equations. You will study Cramer’s Rule in the next
section.
Elementary Linear Algebra: Section 3.3, p.124
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3.4 Linear Algebra Applied

Planetary Orbits
According to Kepler’s First Law of Planetary Motion,
the orbits of the planets are ellipses, with the sun at
one focus of the ellipse. The general equation of a
conic section (such as an ellipse) is
ax2 + bxy + cy2 + dx + ey+ f = 0.
To determine the equation of the orbit of a planet, an
astronomer can find the coordinates of the planet
along its orbit at five different points (xi, yi) where i =
1, 2, 3, 4, and 5, and then use the determinant
x
2
2
xy
x1 x1 y1
y
2
x
y
1
2
y1 x1 y1 1
2
2
2
2
2
2
2
2
x2 x2 y2 y2 x2 y2 1
x3 x3 y 3 y 3 x3 y 3 1
x4 x4 y4 y4 x4 y4 1
Elementary Linear Algebra: Section 3.4, p.135
x5 x5 y 5 y 5 x5 y 5 1
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