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 Introduction to Eigenvalues
3.5 Application of Determinants
Elementary Linear Algebra
R. Larsen et al. (6 Edition)
投影片設計編製者
淡江大學 電機系 翁慶昌 教授
3.1 The Determinant of a Matrix

the determinant of a 2 × 2 matrix:
 a11 a12 
A 

a
a
 21 22 
 det(A)  | A |  a11a22

 a21a12
Note:
a 11 a 12
 a11 a12 

a

a 21 a 22
 21 a22 
Elementary Linear Algebra: Section 3.1, p.123
2/62

Ex. 1: (The determinant of a matrix of order 2)
2 3
1

2
 2(2)  1(3)  4  3  7
2 1
4 2
 2(2)  4(1)  4  4  0
0 3
2 4
 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.123
3/62

Minor of the entry aij :
The determinant of the matrix determined by deleting the ith row
and jth column of A
a11

a12 
a( i 1)1
M ij 
a( i 1)1

a n1

a1( j 1)

 a( i 1)( j 1)
 a( i 1)( j 1)

a1( j 1)


a1n
a( i 1)( j 1)  a( i 1) n
a( i 1)( j 1)  a( i 1) n


an ( j 1)
an ( j 1)


ann
Cofactor of aij :
Cij  ( 1)i  j M ij
Elementary Linear Algebra: Section 3.1, p.124
4/62

Ex:
 a11 a12
A  a21 a22

a31 a32
a13 
a23 

a33 
a12
 M 21 
a32
a13
a33
 C21  ( 1) 21 M 21   M 21
Elementary Linear Algebra: Section 3.1, p.124
M 22
a11 a13

a31 a33
C22  ( 1) 22 M 22  M 22
5/62

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.124
6/62

Ex 2: Find all the minors and cofactors of A.
0 2 1 
A  3  1 2
4 0 1
Sol: (1) All the minors of A.
1 2
3 2
3 1
 M 11 
 1, M 12 
4
 5, M 13 
0 1
4 0
4 1
M 21 
2 1
0 1
 2,
2 1
M 31 
 5,
1 2
Elementary Linear Algebra: Section 3.1, p.125
0 2
0 1
 8
M 22 
 4, M 23 
4 1
4 1
0 2
0 1
 6
M 32
 3, M 33 
3 1
3 2
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Sol: (2) All the cofactors of A.
 Cij  (1)i  j M ij
 C11  
3 1
4
 5, C13  
 1, C12  
4 0
4 1
1
1 2
0
3 2
0 1
0 2
 4, C23  
C21  
 2, C22  
8
4
1
0 1
4 1
2 1
0 1
2 1
 3,
C31  
 5, C 32  
3 2
1 2
Elementary Linear Algebra: Section 3.1, p.125
C33  
0
2
3 1
 6
8/62

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 |  aijCij  ai1Ci1  ai 2Ci 2    ainCin
j 1
(Cofactor expansion along the i-th row, i=1, 2,…, n )
or
n
(b) det( A) | A |  aij Cij  a1 j C1 j  a2 j C2 j    anj Cnj
i 1
(Cofactor expansion along the j-th row, j=1, 2,…, n )
Elementary Linear Algebra: Section 3.1, p.126
9/62

Ex: The determinant of a matrix of order 3
 a11 a12 a13 
A  a21 a22 a23 


a31 a32 a33 
 det( A)  a11C11  a12C12  a13C13
 a21C21  a22C22  a23C23
 a31C31  a32C32  a33C33
 a11C11  a21C21  a31C31
 a12C12  a22C22  a32C32
 a13C13  a23C23  a33C33
Elementary Linear Algebra: Section 3.1, Addition
10/62

Ex 3: The determinant of a matrix of order 3
0 2 1 
A  3  1 2
4 0 1
Ex 2
 C11  1, C12  5, C13  4
C21  2, C22  4, C23  8
C31  5,
C 32  3, C33  6
Sol:
 det(A)  a11C11  a12 C12  a13 C13  (0)(1)  (2)(5)  (1)(4)  14
 a21C21  a22 C22  a23C23  (3)(2)  (1)(4)  (2)(8)  14
 a31C31  a32 C32  a33 C33  (4)(5)  (0)(3)  (1)(6)  14
 a11C11  a21C21  a31C31  (0)(1)  (3)(2)  (4)(5)  14
 a12 C12  a22 C22  a32 C32  (2)(5)  (1)(4)  (0)(3)  14
 a13C13  a23 C23  a33C33  (1)(4)  (2)(8)  (1)(6)  14
Elementary Linear Algebra: Section 3.1, p.126
11/62

Ex 5: (The determinant of a matrix of order 3)
0 2 1 
A  3  1 2  det( A)  ?
4  4 1 
Sol:
2
11  1 2
1 2 3
C11  (1)
 1 C12  (1)
 (1)(5)  5
0 1
4 1
1 3
C13  (1)
3 1
4
4 0
 det(A)  a11C11  a12C12  a13C13
 (0)(1)  (2)(5)  (1)(4)
 14
Elementary Linear Algebra: Section 3.1, p.128
12/62


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 2
 1 1
A
0
2

4
3
3 0
0 2 
 det( A)  ?
0 3

0  2
Elementary Linear Algebra: Section 3.1, p.127
13/62
Sol:
det(A)  (3)(C13 )  (0)(C23 )  (0)(C33 )  (0)(C43 )
 3C13
1 1
 3(1)13 0
3
2
2 3
4 2
2
2

2 1 1
22  1
23  1 1 
 3(0)( 1)
 (2)( 1)
 (3)( 1)

4

2
3

2
3
4


 30  (2)(1)( 4)  (3)( 1)( 7)
 (3)(13)
 39
Elementary Linear Algebra: Section 3.1, p.127
14/62

The determinant of a matrix of order 3:
Subtract these three products.
 a11 a12 a13 
A  a 21 a 22 a 23
 a 31 a 32 a 33 
a11
a12
a13
a11
a21 a22
a23
a21 a22
a31
a33
a31
a32
a12
a32
Add these three products.
 det( A) | A | a11a22 a33  a12 a23a31  a13a21a32  a31a22 a13
 a32 a 23 a11  a33a21a12
Elementary Linear Algebra: Section 3.1, p.127
15/62

Ex 5:
–4 0 6
0 2 1 0 2
A   3  1 2 3  1


4  4 1 4  4
0 16 –12
 det(A) | A | 0  16  12  (4)  0  6  2
Elementary Linear Algebra: Section 3.1, p.128
16/62

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.128
17/62

Ex:
a11 a12 a13 
 0 a22 a23 
0 0 a 
33 

 a11 0 0 
 a21 a22 0 
a a a 
 31 32 33 
a11 0 0 
 0 a22 0 
0 0 a 
33 

upper triangular
lower triangular
diagonal
Elementary Linear Algebra: Section 3.1, p.128
18/62

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
det( A) | A | a11a22 a33 ann
Elementary Linear Algebra: Section 3.1, p.129
19/62

Ex 6: Find the determinants of the following triangular matrices.
0
2
 4 2
(a) A  
 5 6
1
5

0 0
0 0

1 0
3 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.129
20/62
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
21/62
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( k ) ( A)  det(B)  k det(A)
(i.e. ri( k ) ( A)  k A )
(c) B  rij( k ) ( A)  det(B)  det(A)
(i.e. rij( k ) ( A)  A )
Elementary Linear Algebra: Section 3.2, p.134
22/62

Ex:
1
A  0

1
4
A1  0
1
2 3
det(A)  2
1 4

2 1
8 12
2
3
0 1 4 
1
1 4  A2  1 2 3 A3   2  3  2
 1
1 2 1 
2 1 
2
1 
A1  r1( 4) ( A)  det(A1)  det(r1( 4) ( A))  4 det(A)  (4)(2)  8
A2  r12 ( A)
 det(A2)  det(r12 ( A))   det(A)  (2)  2
A3  r12( 2) ( A)  det(A3)  det(r12( 2) ( A))  det(A)  2
Elementary Linear Algebra: Section 3.2, Addition
23/62

Notes:
det(rij ( A))   det(A)
 det(A)   det(rij ( A))
1
det( ri ( A))  k det( A)  det( A)  det( ri( k ) ( A))
k
(k )
det(rij(k ) ( A))  det(A)
 det(A)  det(rij(k ) ( A))
Elementary Linear Algebra: Section 3.2, pp.134-135
24/62
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  det( A)  ?
0 1  3
Sol:
2  3 10
det(A)  1
0
2
1
r12
1
2
2
 2   2  3 10
3
0 1 3
Elementary Linear Algebra: Section 3.2, pp.134-135
25/62
r12( 2 )
1
2
2
( 1 )
r2 7
  0  7 14  (1)(
0 1 3
( 1)
r23
1
1
7
1 2 2
) 0 1 2
0 1 3
1 2 2
 7 0 1  2  (7)(1)(1)(1)  7
0 0 1
Elementary Linear Algebra: Section 3.2, pp.134-135
26/62

Notes:
EA  E A
(1) E  Rij
 E  Rij  1
 EA  rij  A   A  Rij A  E A
(k )
(2) E  Ri( k )  E  Ri  k
 EA  ri( k )  A  k A  Ri( k ) A  E A
(k )
(3) E  Rij( k )  E  Rij  1
 EA  rij( k )  A  1 A  Rij( k ) A  E A
Elementary Linear Algebra: Section 3.2, Addition
27/62

Determinants and elementary column operations

Thm: (Elementary column operations and determinants)
Let A and B be square matrices.
(a) B  cij ( A)
 det( B)   det( A)
(i.e. cij ( A)   A )
(b) B  ci( k ) ( A)  det(B)  k det(A)
(i.e. ci( k ) ( A)  k A )
(c) B  cij( k ) ( A)  det(B)  det(A)
(i.e. cij( k ) ( A)  A )
Elementary Linear Algebra: Section 3.2, p.135
28/62

Ex:
2 1  3
A  4 0 1 
0 0 2 
det(A)  8
1 1  3
A1  2 0 1 
0 0 2 
1 2  3
A2  0 4 1 
0 0 2 
2 1 0
A3  4 0 1 
0 0 2
1
1
A1  c ( A)  det( A1)  det( c1 ( A))  det( A)  ( )( 8)  4
2
2
A2  c12 ( A)  det(A2)  det(c12 ( A))   det(A)  (8)  8
1
( )
2
1
( 4)
A3  c23( 3) ( A)  det(A3)  det(c23( 3) ( A))  det(A)  8
Elementary Linear Algebra: Section 3.2, p.135
29/62

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.136
30/62

Ex:
1 2 3
1 4 0
1 1 1
0 0 0 0
4 5 6
2 5 0 0
3 6 0
2 2 2 0
4 5 6
1 4 2
1 5 2 0
1 6 2
1
2
3
4
5
6 0
2 4 6
Elementary Linear Algebra: Section 3.2, Addition
1
8
4
2 10 5  0
3 12 6
31/62

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.138
32/62

Ex 5: (Evaluating a determinant)
2
 3 5
A   2  4  1


6 
 3 0
3 5
2
Sol:
3 5 4
det(A)  2  4  1  2  4 3
3 0
6
3 0
0
( 2)
C13
5 4
 (3)(1)
 (3)(1)  3
4 3
3 5
2 ( 54 )  3 5 2
r12
det(A)  2  4  1  52 0 53
3 0
6
3 0 6
31
2
5
3
5
3
 (5)(1)
 (5)( )  3
3 6
5
1 2
Elementary Linear Algebra: Section 3.2, p.138
33/62

Ex 6: (Evaluating a determinant)
 2 0 1
 2 1 3
A   1 0 1
 3 1 2
 1 1 3
3  2
2  1
2 3
4  3
2 0
Sol:
2 0 1
2 1 3
det(A)  1 0  1
3 1 2
1 1 3
2
 (1)(1)2 2 1
1
3
3 2
2
2 1
2
2 3 (1) 1
4  3 rr24( 1) 1
2 0 25 3
1
1
5
0
3
2
6
0
Elementary Linear Algebra: Section 3.2, p.139
0 1
1 3
0 1
0 5
0 0
3 2
2 1
2 3
6 4
0 1
2
3
4
1
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8
1
( 3 )
C41
 8 1

13 5
0
0
1 3
 5(1)
3 2
8
1 3
0
0 5
2 3
 (1)(1) 44  8  1 2 (1)  8  1 2
r21
6 4
13 5 6 13 5 6
0 1
 8 1
13
5
 (5)( 27)
 135
Elementary Linear Algebra: Section 3.2, p.140
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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)
a11
a12
a22  b22
a31
a22  b22
a32
a13
a11
a12
a13
a11
a12
a13
a23  b23  a21 a22 a23  b21 b22 b23
a31 a32 a33 a31 a32 a33
a33
Elementary Linear Algebra: Section 3.3, p.143
36/62

Ex 1: (The determinant of a matrix product)
1  2 2 
A  0 3 2
1 0 1 
1 
2 0
B  0  1  2
3 1  2
Find |A|, |B|, and |AB|
Sol:
1 2 2
| A | 0
1
3
0
2  7
1
Elementary Linear Algebra: Section 3.3, p.143
2
0
1
| B | 0  1  2  11
3 1 2
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1  8 4
1 
1  2 2 2 0
AB  0 3 2 0  1  2  6  1  10
1 0 1 3 1  2 5 1
 1 
8
4
1
| AB | 6  1  10  77
5 1
1

Check:
|AB| = |A| |B|
Elementary Linear Algebra: Section 3.3, p.143
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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  20 40
A   30
0
50,


 20  30 10 
Find |A|.
1
2 4
3
0 5 5
2 3 1
Sol:
1 2 4
 1  2 4
3



A

10
3
0 5  (1000)(5)  5000
A  10 3
0 5


2 3 1
 2  3 1
Elementary Linear Algebra: Section 3.3, p.144
39/62

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 2  1
A  3  2 1 
3 2  1
0 2  1
B  3  2 1 
3 2
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.145
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Thm 3.8: (Determinant of an inverse matrix)
1
If A is invertible, then det( A 1 ) 
.
det( A)
 Thm 3.9: (Determinant of a transpose)

If A is a square matrix, then det( AT )  det( A).

Ex 4:
Sol:
1 0 3 
A  0  1 2
2 1 0
1
0
3
 | A | 0  1 2  4
2 1 0
Elementary Linear Algebra: Section 3.3, pp.146-148
1
(a) A  ?

(b)
AT  ?
1 1

A 4
AT  A  4
A1 
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
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.147
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
Ex 5: Which of the following system has a unique solution?
 x3
 x3
 x3
 1

4
 4
2 x2

x3

1
3x1
 2 x2

x3

4
3x1
 2 x2

x3
 4
(a)
3x1
3x1
(b)
2 x2
 2 x2
 2 x2
Elementary Linear Algebra: Section 3.3, p.148
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Sol:
(a) Ax  b

A 0

This system does not have a unique solution.
(b) Bx  b


B  12  0
This system has a unique solution.
Elementary Linear Algebra: Section 3.3, p.148
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3.4 Introduction to Eigenvalues

Eigenvalue problem:
If A is an nn matrix, do there exist n1 nonzero matrices x
such that Ax is a scalar multiple of x？

Eigenvalue and eigenvector:
A：an nn matrix
：a scalar
x： a n1 nonzero column matrix
Eigenvalue
Ax  x
(The fundamental equation for
the eigenvalue problem)
Eigenvector
Elementary Linear Algebra: Section 3.4, p.152
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
Ex 1: (Verifying eigenvalues and eigenvectors)
1 4 
A

 2 3
1
x1   
1
2
x2   
 1
Eigenvalue
1 4 1 5
1
Ax1  
    5   5 x1



2 3 1 5
1
Eigenvector
Eigenvalue
1 4   2    2 
2
Ax 2  
    1   (1) x2



2 3  1  1 
 1
Eigenvector
Elementary Linear Algebra: Section 3.4, p.152
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

Question:
Given an nn matrix A, how can you find the eigenvalues and
corresponding eigenvectors?
Note:
Ax  x  (I  A) x  0
(homogeneous system)
If (I  A) x  0 has nonzero solutions iff det(I  A)  0.

Characteristic equation of AMnn:
det( I  A)  (I  A)  n  cn1n1    c1  c0  0
Elementary Linear Algebra: section 3.4, p.153
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
Ex 2: (Finding eigenvalues and eigenvectors)
1 4 
A

2
3


Sol: Characteristic equation:
 1  4
I  A 
2  3
 2  4  5  (  5)(  1)  0
   5,  1
Eigenvalues: 1  5, 2  1
Elementary Linear Algebra: Section 3.4, p.153
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(1)1  5
(2)2  1
 4  4  x1  0
 (1I  A) x  
 



  2 2   x2   0 
 x1  t  1
 x   t   t 1, t  0
 2    
 2  4  x1  0
 (2 I  A) x  
 



  2  4   x2   0 
 x1   2t   2 
 x    t   t  1, t  0
 2    
Elementary Linear Algebra: Section 3.4, p.153
49/62

Ex 3: (Finding eigenvalues and eigenvectors)
 1 2  2
A 1 2
1


 1  1 0 
Sol: Characteristic equation:
 1
I  A   1
1
2
2
  2  1  (  1)(  1)(  3)  0
1

T heeigenvalues : 1  1, 2  1, 3  3
Elementary Linear Algebra: Section 3.4, p.154
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 0  2 2  1 0 2 
 1I  A   1  1  1 ~ 0 1  1

 

1
1  0 0 0 
 1
1  1
 x1   2t 
  2
 x    t   eigenvect ors : t  1 , t  0
 2 

 
 x3   t 
 1 
 2  2 2  1 0  2
 2 I  A    1  3  1 ~ 0 1 1 

 

1  1 0 0 0 
 1
 x1   2t 
2
 x    t   eigenvectors : t  1, t  0
 2  
 
 x3   t 
 1 
2  1
Elementary Linear Algebra: Section 3.4, p.155
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3  3
 2  2 2  1 0 2
 3I  A   1 1  1 ~ 0 1 1

 

1
3  0 0 0
 1
 x1   2t 
 x     t   eigenvectors :
 2 

 x3   t 
Elementary Linear Algebra: Section 3.4, p.156
  2
t   1 , t  0
 1 
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3.5 Applications of Determinants

Matrix of cofactors of A:
C11 C12  C1n 
C

C

C
2n 
Cij   21 22
 

 


Cn1 Cn 2  Cnn 

Cij  (1)i  j Mij
Adjoint matrix of A:
adj( A)  Cij 
T
C11
C
  12
 

C1n
Elementary Linear Algebra: Section 3.5, p.158
C21  Cn1 
C22  Cn 2 

 

C2 n  Cnn 
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
Thm 3.10: (The inverse of a matrix given by its adjoint)
If A is an n × n invertible matrix, then
1
A 
adj( A)
det(A)
1

Ex:
a b 
A

c
d


 det( A)  ad  bc
 d  b
adj( A)  


c
a


Elementary Linear Algebra: Section 3.5, p.159
1
A 
adj( A)
det  A
1  d  b

ad  bc   c a 
1
54/62

Ex 1 & Ex 2:
2
 1 3
A   0  2 1 
 1
0  2
(a) Find the adjoint of A.
(b) Use the adjoint of A to find A1
i j

C

(

1
)
M ij
Sol:
ij
2 1
0 1
0 2
 C11  
 4, C12  
 1, C13  
2
0 2
1 2
1 0
C21  
C31  
3 2
0 2
3
 6,
2
2 1
 7,
Elementary Linear Algebra: Section 3.5, p.160
C22  
C 32  
1 2
1
2
 0, C23  
1 3
1
0
3
1 3
2
 1, C33  
1
0 2
1 2
0
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 cofactor matrix of A  adjoint matrix of A
 4 1 2
Cij  6 0 3
7 1 2
 


adj( A)  Cij 
inverse matrix of A
1
A1 
adj( A)
det  A
4 6 7  43 2
 13 1 0 1   13 0

 
2 3 2  23 1
Check:
T
4 6 7
 1 0 1 


2 3 2
 det  A  3

1
3

2

3
7
3
AA 1  I
Elementary Linear Algebra: Section 3.5, p.160
56/62

Thm 3.11: (Cramer’s Rule)
a11 x1  a12 x2    a1n xn  b1
a21 x1  a22 x2    a2 n xn  b2

an1 x1  an 2 x2    ann xn  bn
A  aij nn  A(1) , A( 2) ,, A( n ) 
Ax  b
det(A) 
 a1n
a11
a12
a21
a22  a2 n


an1
an 2
 b1 
 x1 
b 
x 
2
x   2 b   


b 
x 
 n
 n
0

(this system has a unique solution)
 ann
Elementary Linear Algebra: Section 3.5, p.163
57/62

Aj  A(1) , A( 2) ,, A( j 1) , b, A( j 1) ,, A(n)

 a11  a1( j 1) b1 a1( j 1)  a1n 
a


a
b
a

a
21
2 ( j 1)
2
2 ( j 1)
2n



 
 
a


a
b
a

a
n ( j 1)
n
n ( j 1)
nn 
 n1
( i.e. det( Aj )  b1C1 j  b2C2 j    bnCnj )
det( A j )
 xj 
,
det( A)
j  1, 2 ,, n
Elementary Linear Algebra: Section 3.5, p.163
58/62

Pf:
det(A)  0
A x = b,
1
x A b 
adj( A)b
det( A)
C11 C21  Cn1   b1 

 
1 C12 C22  Cn 2  b2 


   
det (A)  

 
C
C

C
2n
nn  bn 
 1n
1
b1C11

1 b1C12

det( A) 
b C
 1 1n
 b2C21
 b2C22

 b2C2 n
Elementary Linear Algebra: Section 3.5, p.163
   bn Cn1 
   bnCn 2 



   bn Cnn 
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 xj 

1
(b1C1 j  b2C2 j    bnCnj )
det(A)
det(A j )
det(A)
j  1,2 ,, n
Elementary Linear Algebra: Section 3.5, p.163
60/62

Ex 4: Use Cramer’s rule to solve the system of linear equations.
 x  2 y  3z  1
2x

z  0
3x  4 y  4 z  2
Sol:
1
det (A)  2
3
2
0
4
3
1  10
4
1
det( A1 )  0 0
2 4
1
1 1  3
det( A2 )  2
3
0
2
det( A1 ) 4
x

det( A) 5
1  15,
4
det( A3 )  2
3
det( A2 )  3
y

det( A)
2
Elementary Linear Algebra: Section 3.5, p.163
3
2
2
1 8
4
1
0 0  16
4 2
det( A3 )  8
z

det( A)
5
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Keywords in Section 3.5:

matrix of cofactors : 餘因子矩陣

adjoint matrix : 伴隨矩陣

Cramer’s rule : Cramer 法則
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