資訊科學數學14 :
Determinants & Inverses
陳光琦助理教授 (Kuang-Chi Chen)
chichen6@mail.tcu.edu.tw
1
Linear Equations and Matrices
Determinants
2
3.1 Determinants
• With each nn matrix A it is possible to associate a
scalar det(A), called the determinant of the matrix,
whose value will tell us whether the matrix is
singular or not.
• Case 1: 11 matrices
- If A = (a), then A will have a multiplicative inverse
iff a≠0 .
- A is nonsingular iff det(A)≠0 .
3
22 Matrices
• Case 2: 22 matrices
 a11 a12 
- Let A = 
.

 a21 a22 
- A will be nonsingular iff det(A) = a11a22 – a12a21≠ 0 .
4
33 Matrices
• Case 3: 33 matrices
a
a
a

11
12
- Let A =
. 13 
a

a
a
21
22
23 

 a31 a32 a33 
- A will be nonsingular iff
det(A) = a11a22a33 + a12a31a23 + a13a21a32 – a11a32a23 –
a12a21a33 – a13a31a22 ≠ 0 .
5
Example 4 & 5
• Example 4
If A = [a11] is a 11 matrix, then det(A) = a11 .
• Example 5
If
 a11 a12 
A
 ⇒ det(A) = a a – a a
a
a
11 22
12 21
 21 22 
2  3 ⇒ det(A) = (2)(5) – (-3)(4) = 22
A

4
5


6
Example 6 & 7
• Example 6
If  a11 a12 a13 
A  a21 a22 a23  ⇒ det(A) = a11a22a33 + a12a31a23 + a13a21a32
a31 a32 a33 
– a11a32a23 – a12a21a33 – a13a31a22
• Example 7
1 2 3 
A  2 1 3 ⇒ det(A) = (1)(1)(2) + (3)(2)(1) + (2)(3)(3)
– (3)(1)(3) – (1)(1)(3) – (2)(2)(2) = 6
3 1 2
If
7
Properties of Determinants
• Theorem 3.1
The determinants of a matrix and its transpose are
equal, i.e., det(A) = det(AT).
8
Example 8
• Example 8
If
1 2 3 
A  2 1 3
3 1 2
1 2 3 
 AT  2 1 1


3 3 2
⇒ det(AT) = (1)(1)(2) + (3)(1)(2) + (2)(3)(3)
– (3)(1)(3) – (1)(1)(3) – (2)(2)(2)
= 6 = det(A)
9
Theorem 3.2 & 3.3
• Theorem 3.2
If matrix B results from matrix A by interchanging
two rows (or two columns) of A, then
det(B) = -det(A).
• Theorem 3.3
If two rows (or columns) of A are equal, then
det(A) = 0.
10
Example 9 & 10
• Example 9
If
2 1
3 2
 7 and
 7
3 2
2 1
• Example 10
If
1
2 3
1 0 7  0
1 2 3
11
Theorem 3.4
• Theorem 3.4
If a row (or column) of A consists entirely of zeros,
then det(A) = 0.
• Example 11
1 2 3
4 5 6 0
0 0 0
12
Theorem 3.5
• Theorem 3.5
If B is obtained from A by multiplying a row (column)
of A by a real number c, then
det(B) = c det(A) .
• Example 12
2 6
1 3
1 1
2
 23
 64  1  18
1 12
1 12
1 4
13
Example 13
• Example 13
1 2 3
1 2 3
1 2 1
1 5 3  21 5 3  231 5 1  230  0
2 8 6
1 4 3
1 4 1
14
Theorem 3.6
• Theorem 3.6
If B = [bij] is obtained from A = [aij] by adding to
each element of the rth row (column) of A a constant
c times the corresponding element of the sth row
(column) r≠s of A, then det(B) = det(A) .
• Example 14 1 2 3
5 0 9
2 1 3  2 1 3
1
0
1
1
0
1
15
Theorem 3.7
• Theorem 3.7
If a matrix A = [aij] is upper (lower) triangular, then,
then det(A) = a11 a22 … ann .
• Corollary 1.3
The determinant of a diagonal matrix is the product
of the entries on its main diagonal.
16
Example 15
• Example 15
2 3 4
3 0 0 
 5 0 0 
A  0 4 5  , B   2 5 0  , C   0 4 0 
0 0 3
6 8 4
 0 0 6 
det( A)  24, det( B)  60, det(C )  120
17
Elementary Operations
• Elementary row and elementary column operations
I - Interchange rows (columns) i and j :
ri ⇔ rj (ci ⇔ cj )
II - Replace row (column) i by a nonzero value k times
row (column) i :
kri ⇔ ri (kci ⇔ ci )
III - Replace row (column) j by a nonzero value k times
row (column) i+ row (column) j :
kri + rj ⇔ rj (kci + cj ⇔ cj )
18
… then …
det( Ari  rj )   det( A), i  j
det(Akri ri )  k det( A)
det (Akri  rj rj )  det( A), i  j
19
Example 16
• E.g. 16
 4 3 2
A   3  2 5


2 4 6
4 3 2
det( A)  2 det( A1 r3 r3 )  2 det (  3 2 5  )
2
1 2 3 
4 3 2
1 2 3 
 2 det(  3 2 5 
)  ( 1) 2 det(  3 2 5  )
1 2 3  r  r
 4 3 2 
1
3
20
Example 16
(cont’d)
3 
1 2
1 2 3 
)  2 det( 0 -8 4  )
 2 det(  3 2 5 
0 -5 10 
 4 3 2  -3r1  r2 r2
4 r  r  r
1
3
3
3 
3 
1 2
1 2




)  2 det( 0 -8 4  )
 2 det( 0 -8 4 
0 0  304 
0 -5 10   5 r  r r
3
8 2 3
det( A)  2(1)(8)(30 / 4)  120
21
Theorem 3.8
• Theorem 3.8
The determinant of a product of two matrices is the
product of their determinants det(AB) = det(A)det(B) .
• Example 17
1 2
A

3
4


A  2
 4 3
AB  

10
5


2  1
B

1
2


B 5
AB  10  A B
22
Example 17
(cont’d)
• Remark
 1 0 
BA  

7
10


AB≠BA
|BA| = |B| |A|= -10 = |AB|
23
Corollary 3.2
• Corollary 3.2
If A is nonsingular, then det(A) ≠ 0,
thus det(A-1) = 1/det(A).
If A is singular, then det(A) = 0
( 1 = |I| = |AA-1| = |A| |A-1| )
24
• Example 18
Example 18
1 2
A

3
4


1 
 2
A 

3/
2

1/
2


1
det( A)  2
1
1
det( A )   
2 det( A)
1
25
Cofactor Expression and
Applications
26
3.2 Cofactor Expression and Applications
Cofactor expression and applications
• Definition – Minor and cofactor
Let A = [aij] be an nn matrix. Let Mij be the (n-1)
(n-1) submatrix of A obtained by deleting the ith row
and jth column of A. The determinant det(Mij) is
called the minor of aij. The cofactor Aij of aij is
defined as
Aij  (1)i  j det (M ij )
27
Example 1
• E.g. 1
Let
 3  1 2
A   4 5 6


7 1 2
  
  
  
4 6
det( M 12 ) 
 8  42  34
7 2
A12  (1)1 2 det( M 12 )  (1)(34)  34
3 1
det( M 23 ) 
 3  7  10
7 1
23
A23  (1)
1 2
det( M 31 ) 
 6  10  16
5 6
det( M 23 )  (1)(10)  10
A31  (1)31 det( M 31 )  (1)(16)  16
28
Theorem 3.9
• Theorem 3.9
Let A = [aij] be an nn matrix. Then
for each 1≤ i ≤ n,
det(A) = ai1Ai1 + ai2Ai2 + … + ainAin , and
for each 1≤ j ≤ n,
det(A) = a1jA1j + a2jA2j + … + anjAnj .
29
Example 2
To evaluate the determinant
1
2 3
4
4 2 1
3
3 0 0 3
2 0 2 3
1
4
3
2
2 3 4
2 3 4
1 3 4
2 1 3
 (1)31 (3) 2 1 3  (1)3 2 (0) 4 1 3
0 0 3
0 2 3
2 2 3
0 2 3
1 2 4
1 2 3
 (1)33 (0) -4 2 3  ( 1)3 4 (3) 4 2 1
2 0 3
2 0 2
 (1)(3)(20)  0  0  (1)(3)(4)  48
30
Example 3
Consider the determinant of the matrix
1
4
3
2
2 3 4
1
2 1 3
4

0 0 3
3
0 2 3 c  c c
2
4
1
2 3 5
2 1 1
0 0 0
0 2 5
4
2 3 5
0 4 6
 (1)31 (3) 2 1 1
 ( 1) 4 (3) 2 1 1
0 2 5 r  r r
0 2 5
1
2
1
 (1) 4 (3)(2)(8)  48
31
Theorem 3.10
• Theorem 3.10
If A = [aij] be an nn matrix, then
ai1Ak1 + ai2Ak2 + … + ainAkn = 0, for i≠k ,
a1jA1k + a2jA2k + … + anjAnk = 0, for j≠k .
32
Example 4
• E.g. 4
1 2 3
A   2 3 1 
 4 5  2
3
2 1 2
A21   1
 19
5 2
A22   1
1 3
 14
4 2
A23   1
1 2
3
4 5
2 2
23
a31 A21  a32 A22  a33 A23  419  5 14   23  0
a11 A21  a12 A22  a13 A23  119  2 14  33  0
33
Adjoint
• Definition – Adjoint
Let A = [aij] be an nn matrix. The nn matrix adj A,
called the adjoint of A, is the matrix whose j, ith
element is the cofactor Aij of aij . Thus
 A11
A
adjA   12
 

 A1n
A21  An1 
A22  An 2 

 

A2 n  Ann 
34
Remark
• Remark
The adjoint of A is formed by taking the
transpose of the matrix of cofactors Aij of the
elements of A.
35
Example 5
• Example 5
Compute adj A
3  2 1 
A  5 6
2 
1 0  3
36
Solution
A11   1
11
6
2
0 3
 18
A12   1
5 2
 17
1 3
A13   1
5 6
1 2
1 3
A21   1
1 0
2 1
A22   1
2 2
A23   1
23
 6
1
1 3
A32   1
3 1
 1
5 2
A33   1
3 2
3 2
3 3
2 1
 6
0 3
3
A31   1
2 1
 10
6 2
31
 10
3 2
 2
1 0
5
6
 28
 18  6  10
Then, adj A   17  10  1 
  6  2
28 
37
Theorem 3.11
• Theorem 3.11
If A = [aij] be an nn matrix, then
A(adj A) = (adj A)A = det(A) In .
38
• E.g. 6
Example 6
Consider the matrix
3  2 1 
A  5 6
2 
1 0  3
0 
3 2 1   18 6 10   94 0
1 0 0 
5 6 2   17 10 1    0 94 0   94 0 1 0 


 



1 0 3  6 2 28   0
0 0 1 
0 94 
and
 18 6 10  3 2 1 
1 0 0 
 17 10 1  5 6 2   94 0 1 0 





 6 2 28  1 0 3
0 0 1 
39
• Corollary 3.3
Corollary 3.3
If A = [aij] be an nn matrix and det(A)≠0, then
 A
 det  A

A

1
1
adjA   det  A
A 
det  A
 
 A

 det  A
A
A 

det  A
det  A

A
A 

det  A
det  A

 
A
A 


det  A
det  A
40
Example 7
• Example 7
Consider the matrix
3 2 1 
A  5 6 2 
1 0 3
Then det(A) = -94, and
 18
94
1

1
17
A 
adjA


  94
det( A)
 946
6
94
10
94
2
94

1 
94 
128 
94 
10
94
41
Theorem 3.12
• Theorem 3.12
A matrix A = [aij] is nonsingular iff det(A) ≠ 0.
• Corollary 3.4
For an nn matrix A, the homogeneous system Ax
= 0 has a nontrival solution iff det(A) = 0.
42
• Example 8
Example 8
Let A be a 4x4 matrix with det(A) = -2
(a) describe the set of all solutions to the
homogeneous system Ax = 0.
(b) If A is transformed to reduced row echelon form B,
what is B?
(c) Given an expression for a solution to the linear
system Ax = b, where b = [b1 , b2 , b3 , b4 ]T .
(d) Can the linear system Ax = b have more than one
solution? Explain.
(e) Does A-1 exist?
43
Solutions of Example 8
• Solutions
(a) Since det(A)≠0, Ax = 0 has only the trivial solution.
(b) Since det(A)≠0, A is a nonsingular matrix, so B = In
(c) A solution to the given system is given by x = A-1b
(d) No. The solution is unique.
(e) Yes.
44
Nonsingular Equivalence
• List of nonsingular equivalence
The following statements are equivalent.
1. A is nonsingular.
2. x = 0 is the only solution to Ax = 0.
3. A is row equivalence to In .
4. The linear system Ax = b has a unique solution for
every n1 matrix b.
5. det(A)≠0 .
45
Determinants
•
•
•
•
Linearly independent
Nonsingular
Trivial solution x = 0 to Ax = 0
det(A) ≠ 0
46
Determinants
•
•
•
•
Linearly dependent
Singular
Nontrivial solution to Ax = 0
det(A) = 0
47
Cramer’s Rule
Theorem 3.13 (Cramer’s Rule)
Let a11x1 + a12x2 + … + a1nxn = b1
a21x1 + a22x2 + … + a2nxn = b2
…
an1x1 + an2x2 + … + annxn = bn
Then,
x1 = det(A1)/det(A) , x2 = det(A2)/det(A) , … ,
xn = det(An)/det(A) .
48
Cramer’s Rule
Cramer’s Rule for solving the linear system Ax = b,
where A is nn, is as follows:
Step 1. Compute det(A). If det(A) = 0, Cramer’s rule is
not applicable. Use Gauss-Jordan Reduction.
Step 2. If det(A)≠0, for each i,
xi = det(Ai)/det(A) ,
where Ai is the matrix obtained from A by replacing
the ith column of A by b.
49
Example 9
• Consider the following linear system:
-2x1 + 3x2 – x3 = 1
x1 + 2x2 – x3 = 4
-2x1 – 2x2 + x3 = -3
Then
2
3
1
A  1 2 1  2
2 1 1
50
Example 9
(cont’d)
Hence,
x1 
x2 
1
3
1
4
2
1
3 1
1
A
4

2
2
2
1
1
2
3
1
1
4
1
1
2
4
2 3
A
1
6

3
2
x3 
2 1 3
A
8

4
2
51
Polynomial Interpolation Revisited
• Polynomial Interpolation Revisited
To find a quadratic polynomial that interpolates the
following points:
(x1, y1), (x2, y2), (x3, y3),
where x1≠x2 , x1≠ x3 , x2≠ x3 .
52
… more …
The polynomial has the form:
y = a2x2 + a1x + a0 .
The corresponding linear system
y1 = a2x12 + a1x1 + a0 ,
y2 = a2x22 + a1x2 + a0 ,
y3 = a2x32 + a1x3 + a0 .
53
… more …
The coefficient matrix
 x12
 2
 x2
 x32
x1 1

x2 1
x3 1
The Vandermount determinant
(x1 – x2 )( x1 – x3 )( x2 – x3 )
54
55
Linear Equations and Matrices
LU-Factorization
56
LU-Factorization
• 1.8 LU-Factorization
If a square matrix can be reduced to upper
triangular form using only 3 row operations,
then it is possible to represent the reduction
process in terms of a matrix factorization.
57
Type I Operation
An elementary matrix of type I is a matrix obtained by
interchanging two rows of identity matrix I.
Example
0 1 0 
0 1 0 
E1  1 0 0  , E11  1 0 0 
0 0 1 
0 0 1 
E1 A  Interchange the first row and
the second row of A
AE1  Interchange the first column
and the second column of A
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Type II Operation
An elementary matrix of type II is a matrix obtained
by multiplying a row of I by a nonzero constant.
Example
1 0 0 
1 0 0 
E2  0 1 0  , E2 1  0 1 0 
0 0 3
0 0 1/ 3
E2 A  Multiply the third row of A by 3
AE2  Multiply the third column of A by 3
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Type III Operation
An elementary matrix of type III is a matrix obtained
from I by adding a multiple of one row to another row.
Example
1 0 3 
1 0 3
E3  0 1 0  , E31  0 1 0 
0 0 1 
0 0 1 
E3 A  Add three times the third row to
the first row of A
AE3  Add three times the third column
to the first column of A
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In General
In general, the elementary matrix by adding m times
of row i to row j
1


0

E3   
0
Row j


0

aji
1






0

 1
 1 

, E3   
0

 m  1


 

0
 0  0  1









1




m  1

 
0  0  1
Column i
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Theorem
• Theorem
If A and B are nonsingular square matrices, then AB
is also nonsingular.
i.e. (AB)-1 = B-1A-1 .
In general, if E1, E2, …, Ek are all nonsingular, then
the product E1E2 … Ek is also nonsingular and
(E1E2 … Ek )-1 = Ek-1 … E2-1 E1-1 .
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Row Equivalence
• A matrix B is row equivalent to A if there exists a
finite sequence E1E2 … Ek of elementary matrices
such that B = Ek … E2 E1A .
63
LU-Factorization
• LU-Factorization
If a square matrix can be reduced to upper
triangular form using only 3 row operations,
then it is possible to represent the reduction
process in terms of a matrix factorization.
64
Example
• Let
 2 4 2
A  1 5 2
4  1 9
2 4
E1 A  1 5
 4 1
 1
E1   1/ 2
 0
2 2 4 2
2    0 3 1 
9   4 1 9 
0 0
1 0  , a21  1/ 2
0 1 
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(cont’d)
2 4
E2 A   0 3
 4 1
1 0
E2   0 1
 2 0
2 2 4 2
2
1    0 3 1  E3 A   0
 0
9   0 9 5 
0
1
0  , a31  2
E3  0
 0
1 
2 2 4 2
1    0 3 1 
9 5   0 0 8 
0 0
1 0  , a32  3
3 1 
4
3
66
(cont’d)
2 4 2
E3 E2 E1 A   0 3 1   U
 0 0 8 
A  ( E3 E2 E1 ) 1U  E11 E21 E31U  LU
1
1
L  E11 E21 E31  
2
0

0 0
  1 0 0  1 0 0 
1 0   0 1 0  0 1 0 

 2 0 1  0 3 1 

0 1
67