CHAPTER 2
MATRICES
2.1
2.2
2.3
2.4
2.5
Operations with Matrices
Properties of Matrix Operations
The Inverse of a Matrix
Elementary Matrices
Applications of Matrix Operations
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 2 Linear Algebra Applied
Flight Crew Scheduling (p.47)
Beam Deflection (p.64)
Information Retrieval (p.58)
Computational Fluid Dynamics (p.79)
Data Encryption (p.87)
2/75
2.1 Operations with Matrices

Matrix:
A  [aij ]mn
 a11 a12 a13  a1n 
a

a
a

a
22
23
2n 
 21
  a31 a32 a33  a3n 
 M mn








am1 am 2 am3  amn 
mn
(i, j)-th entry:
aij
row: m
column: n
size: m×n
Elementary Linear Algebra: Section 2.1, p.40
3/75

i-th row vector
ri  ai1

ai 2  ain 
j-th column vector
 c1 j 
c 
2j


cj 
  
c 
 mj 

row matrix
column matrix
Square matrix: m = n
Elementary Linear Algebra: Section 2.1, p.40
4/75

Diagonal matrix:
d1
0
A  diag (d1 , d 2 ,  , d n )  

0


0
d2

0
0
 0
  M nn
 
 d n 

Trace:
If A  [aij ]nn
Then Tr ( A)  a11  a22    ann
Elementary Linear Algebra: Section 2.1, Addition
5/75

Ex:
1 2 3  r1 
A
 

4 5 6 r2 

r1  1 2 3, r2  4 5 6
1 2 3
 c1 c2
A

 4 5 6
c3 
1 
 2
 3
 c1   , c2   , c3   
 4
5 
6 
Elementary Linear Algebra: Section 2.1, Addition
6/75

Equal matrix:
If A  [aij ]mn , B  [bij ]mn
Then A  B if and onlyif aij  bij 1  i  m, 1  j  n

Ex 1: (Equal matrix)
1 2
A

3 4
a b 
B

c d 
If A  B
T hen a  1, b  2, c  3, d  4
Elementary Linear Algebra: Section 2.1, p.40
7/75

Matrix addition:
If A  [aij ]mn , B  [bij ]mn
Then A  B  [aij ]mn  [bij ]mn  [aij  bij ]mn

Ex 2: (Matrix addition)
 1 2  1 3  1  1 2  3  0 5
 0 1   1 2   0  1 1  2   1 3

 
 
 

 1  1
  3   3 
   
 2  2
 1  1  0 
  3  3   0 

  
 2  2 0
Elementary Linear Algebra: Section 2.1, p.41
8/75

Scalar multiplication:
If A  [aij ]mn , c : scalar
ThencA  [caij ]mn


Matrix subtraction:
A  B  A  (1) B
Ex 3: (Scalar multiplication and matrix subtraction)
 1 2 4
A   3 0  1


 2 1 2
0 0
 2
B   1  4 3


3 2
 1
Find (a) 3A, (b) –B, (c) 3A – B
Elementary Linear Algebra: Section 2.1, p.41
9/75
Sol:
(a)
(b)
(c)
 1 2 4  31 32 34  3 6 12
3A  3 3 0  1  3 3 30 3 1   9 0  3

 
 

6
 2 1 2  32 31 32  6 3
0
0
0 0   2
 2
4  3
 B   1 1  4 3    1



3 2  1  3  2
 1
0 0  1 6 12
 3 6 12  2
3 A  B   9 0  3   1  4 3   10 4  6


 
 
4
6  1
3 2  7 0
 6 3
Elementary Linear Algebra: Section 2.1, p.41
10/75

Matrix multiplication:
If A  [aij ]mn , B  [bij ]n p
Then AB  [aij ]mn [bij ]n p  [cij ]m p
Size of AB
n
where cij   aik bkj  ai1b1 j  ai 2b2 j    ainbnj
k 1
 a11 a12  a1n  b  b
 b1n  
 11

1j
 


 
 


b

b

b
21
2
j
2
n


 ai1 ai 2  ain  



  ci1 ci 2  cij  cin 
 

 


 bn1  bnj  bnn  


an1 an 2  ann  

Notes: (1) A+B = B+A, (2) AB  BA
Elementary Linear Algebra: Section 2.1, p.42 & p.44
11/75

Ex 4: (Find AB)
3
 1
 3


B
A 4 2


 4
0
 5
Sol:
 (1)(3)  (3)(4)
AB  (4)(3)  (2)(4)

 (5)(3)  (0)(4)
2
1
(1)(2)  (3)(1) 
(4)(2)  (2)(1)

(5)(2)  (0)(1) 
 9 1 
 4 6 


 15 10
Elementary Linear Algebra: Section 2.1, p.43
12/75

Matrix form of a system of linear equations:
 a11 x1  a12 x2    a1n xn  b1
 a x  a x  a x  b
 21 1 22 2
2n n
2



am1 x1  am 2 x2    amn xn  bm
m linear equations

 a11
a
 21
 

am1
a12
a22

am 2
 a1n   x1   b1 
 a2 n   x2   b2 


      
   
 amn   xn  bm 
=
=
=
A
x
b
Elementary Linear Algebra: Section 2.1, p.45
Single matrix equation
Ax b
m  n n 1
m 1
13/75

Partitioned matrices:
 a11
A  a21

a31
 a11
A  a21

a31
 a11
A  a21

a31
a12
a13
a22
a32
a23
a33
a12
a13
a22
a32
a23
a33
a12
a13
a22
a32
a23
a33
a14 
 A11

a24  
  A21
a34 
submatrix
A12 
A22 
a14   r1 
a24   r2 
  
a34   r3 
a14 
a24   c1 c2

a34 
Elementary Linear Algebra: Section 2.1, Addition
c3
c4 
14/75

Linear combination of column vectors:
 a11
a
A   21
 
a
 m1
a12
a22

am 2
 x1 
x 
x   2

x 
 n
 a1n 
 a2 n 
  c1 c2  cn 

 
 amn 
 a11 
 a1n 
 a11 x1  a12 x2    a1n xn 
 a12 
a 
a 
a 
 a x  a x  a x 
21
21 1
22 2
2n n
22
  x1    x2      xn  2 n 
 Ax  

  
  
  


a 
a 
a 
a x  a x    a x 
 m1 
 mn 
 m2 
 m1 1 m 2 2
mn n  m1
c1
Elementary Linear Algebra: Section 2.1, p.46
c2
cn
15/75

Ex 7： (Solve a system of linear equations)
x1  2 x2
4 x1  5 x2
7 x1  7 x2
 3x3
 6 x3
 8 x3
 0
 3
 6
(infinitely many solutions)
 1 2 3
 x1 
0 
 1
 2
 3
 A  4 5 6, x   x2 , b   3, c1  4, c2   5, c3  6
7 8 9
 x3 
6
7 
 8
9
 x1  2 x2  3 x3 
 1
 2
 3 0
 Ax  4 x1  5 x2  6 x3   x1 4  x2  5  x3 6   3  b
 7 x1  8 x2  9 x3 
7 
 8
9 6
 1 2
 3 0
 1
 14  1 5  (1) 6   3 (onesolution: x   1, i.e. x1  1, x2  1, x3  1)
7  8
9 6
 1
Elementary Linear Algebra: Section 2.1, p.47
16/75
Key Learning in Section 2.1

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..
17/75
Keywords in Section 2.1

row vector: 列向量

column vector: 行向量

diagonal matrix: 對角矩陣

trace: 跡數

equality of matrices: 相等矩陣

matrix addition: 矩陣相加

scalar multiplication: 純量乘法(純量積)

matrix subtraction: 矩陣相減

matrix multiplication: 矩陣相乘

partitioned matrix: 分割矩陣

linear combination: 線性組合
18/75
2.2 Properties of Matrix Operations

Three basic matrix operators:
(1) matrix addition
(2) scalar multiplication
(3) matrix multiplication

Zero matrix:

Identity matrix of order n:
0mn
Elementary Linear Algebra: Section 2.2, 52-55
In
19/75

Properties of matrix addition and scalar multiplication:
If A, B, C  M mn ,
c, d : scalar
Then (1) A+B = B + A
(2) A + ( B + C ) = ( A + B ) + C
(3) ( cd ) A = c ( dA )
(4) 1A = A
(5) c( A+B ) = cA + cB
(6) ( c+d ) A = cA + dA
Elementary Linear Algebra: Section 2.2, p.52
20/75

Properties of zero matrices:
If A  M mn ,
c : scalar
Then (1) A  0 mn  A
(2) A  ( A)  0 mn
(3) cA  0mn  c  0 or A  0mn

Notes:
(1) 0m×n: the additive identity for the set of all m×n matrices
(2) –A: the additive inverse of A
Elementary Linear Algebra: Section 2.2, p.53
21/75

Properties of matrix multiplication:
(1) A(BC) = (AB)C
(2) A(B+C) = AB + AC
(3) (A+B)C = AC + BC
(4) c (AB) = (cA) B = A (cB)

Properties of identity matrix:
If A  M mn
Then (1) AI n  A
( 2) I m A  A
Elementary Linear Algebra: Section 2.2, p.54 & p.56
22/75

Transpose of a matrix:
 a11
a
If A   21
 
a
 m1
a12
a22

am 2
 a11
a
Then AT   12
 
a
 1n
 a1n 
 a2 n 
  M mn
 

 amn 
a21  am1 
a22  am 2 
  M nm


 
a2 n  amn 
Elementary Linear Algebra: Section 2.2, p.57
23/75

Ex 8: (Find the transpose of the following matrix)
 2
(a) A   
8 
(b)
 1 2 3
A   4 5 6


7 8 9
Sol: (a)
 2
A 
 AT  2
8 
(b)
 1 2 3
1
A  4 5 6  AT  2



7 8 9
3
(c)
1
0
0
A  2 4  AT  


1

 1  1
Elementary Linear Algebra: Section 2.2, p.57
1
0


(c) A  2 4
 1  1
8
4 7
5 8

6 9
2 1
4  1
24/75

Properties of transposes:
(1) ( AT )T  A
(2) ( A  B)T  AT  BT
(3) (cA)T  c( AT )
(4) ( AB)T  BT AT
Elementary Linear Algebra: Section 2.2, p.57
25/75

Symmetric matrix:
A square matrix A is symmetric if A = AT

Skew-symmetric matrix:
A square matrix A is skew-symmetric if AT = –A

Ex:
 1 2 3
If A  a 4 5 is symmetric, find a, b, c?


b c 6
Sol:
 1 2 3
1 a b 
T
A

A
A  a 4 5 AT  2 4 c 



  a  2, b  3, c  5
b c 6
3 5 6
Elementary Linear Algebra: Section 2.2, Addition
26/75

Ex:
 0 1 2
If A  a 0 3 is a skew-symmetric, find a, b, c?


b c 0
Sol:
 0  a  b
 0 1 2
A   a 0 3
 AT    1 0  c 


b c 0
 2  3 0 
A   AT  a  1, b  2, c  3

Note: AAT is symmetric
Pf:
( AAT )T  ( AT )T AT  AAT
 AAT is symmetric
Elementary Linear Algebra: Section 2.2, Addition
27/75


Real number:
ab = ba (Commutative law for multiplication)
Matrix:
AB  BA
mn n p
Three situations:
(1) If m  p, then AB is defined, BA is undefined.
(2) If m  p, m  n, then AB  M mm , BA M nn (Sizes are not the same)
(3) If m  p  n, then AB  M mm , BA M mm
(Sizes are the same, but matrices are not equal)
Elementary Linear Algebra: Section 2.2, Addition
28/75

Ex 4:
Sow that AB and BA are not equal for the matrices.
2  1
 1 3
B
A
and


0
2
2

1




Sol:
5
 1 3 2  1 2
AB  





2

1
0
2
4

4


 

7
2  1  1 3 0
BA  





0
2
2

1
4

2


 


Note:
AB  BA
Elementary Linear Algebra: Section 2.2, p.55
29/75

Real number:
ac  bc , c  0
(Cancellation law)
 ab

Matrix:
AC  BC
C0
(1) If C is invertible, then A = B
(2) If C is not invertible, then A  B (Cancellation is not valid)
Elementary Linear Algebra: Section 2.2, p.55
30/75

Ex 5: (An example in which cancellation is not valid)
Show that AC=BC
 1 3
 2 4
 1  2
A
, B
, C



0
1
2
3

1
2






Sol:
1
AC  
0
2
BC  
2
3  1  2  2




1  1
2   1
4  1  2   2




3  1 2    1
So
AC  BC
But
A B
Elementary Linear Algebra: Section 2.2, p.55
4
2
4
2
31/75
Key Learning in Section 2.2

Use the properties of matrix addition, scalar multiplication,
and zero matrices.

Use the properties of matrix multiplication and the identity
matrix.

Find the transpose of a matrix.
32/75
Keywords in Section 2.2

zero matrix: 零矩陣

identity matrix: 單位矩陣

transpose matrix: 轉置矩陣

symmetric matrix: 對稱矩陣

skew-symmetric matrix: 反對稱矩陣
33/75
2.3 The Inverse of a Matrix

Inverse matrix:
Consider A  M nn
If there exists a matrix B  M nn such that AB  BA  I n ,
Then (1) A is invertible (or nonsingular)
(2) B is the inverse of A

Note:
A matrix that does not have an inverse is called
noninvertible (or singular).
Elementary Linear Algebra: Section 2.3, p.62
34/75

Thm 2.7: (The inverse of a matrix is unique)
If B and C are both inverses of the matrix A, then B = C.
Pf:
AB  I
C ( AB)  CI
(CA) B  C
IB  C
BC
Consequently, the inverse of a matrix is unique.

Notes:
(1) The inverse of A is denoted by A1
(2) AA1  A1 A  I
Elementary Linear Algebra: Section 2.3, pp.62-63
35/75

Find the inverse of a matrix by Gauss-Jordan Elimination:
A

- Jordan Eliminatio n
| I  Gauss

  I | A1 
Ex 2: (Find the inverse of the matrix)
4
 1
A

 1  3
Sol:
AX  I
4  x11
 1
 1  3  x

  21
 x11  4 x21
 x  3 x
 11
21
x12   1 0


x22  0 1
x12  4 x22   1 0


 x12  3x22  0 1
Elementary Linear Algebra: Section 2.3, pp.63-64
36/75

x11  4 x21  1
 x11  3 x21  0
(1)
x12  4 x22  0
 x12  3 x22  1
(2)
( 4 )
1
4

1
r12(1) , r21


(1) 
 
  1 0   3  x11  3, x21  1
 1  3  0
0 1  1
( 4 )
1
4

0
r12(1) , r21


( 2) 
 
  1 0   4  x12  4, x22  1
 1  3  1
0 1  1
Thus
 x11
X A 
 x21
1
x12   3  4
-1

(
AX

I

AA
)



x22   1
1
Elementary Linear Algebra: Section 2.3, pp.63-64
37/75

Note:
 1 4  1 0 Gauss JordanElimination 1 0   3  4
(1) 
(
4 ) 
 1  3  0 1 r
0 1 

,
r
12
21
1
1




A
I
I
A1
If A can’t be row reduced to I, then A is singular.
Elementary Linear Algebra: Section 2.3, p.64
38/75

Ex 3: (Find the inverse of the following matrix)
 1  1 0
A   1 0  1
 6 2 3
Sol:
 1  1 0  1 0 0
A  I    1 0  1  0 1 0
 6 2 3  0 0 1
 1  1 0  1 0 0
r13( 6 )

  0 1  1   1 1 0 
 6 2 3  0 0 1
r12( 1)
 1  1 0  1 0 0
 0 1  1   1 1 0
0 0  1  2 4 1
( 4)
r23
Elementary Linear Algebra: Section 2.3, p.65
 1  1 0  1 0 0
0 1  1   1 1 0
0  4 3  6 0 1
 1  1 0  1 0 0

 0 1  1   1 1 0
0 0 1   2  4  1
r3( 1)
39/75
 1  1 0  1 0 0
 1 0 0   2  3  1
(1 )
r21
 0 1 0   3  3  1  0 1 0   3  3  1
0 0 1   2  4  1
0 0 1   1  4  1
(1)
r32
 [ I  A1 ]
So the matrix A is invertible, and its inverse is
 2  3  1
A1    3  3  1
 2  4  1

Check:
AA 1  A1 A  I
Elementary Linear Algebra: Section 2.3, p.65
40/75

Power of a square matrix:
(1) A0  I
(2) Ak  
AA

A



(k  0)
k factors
(3) Ar  As  Ar s
r, s : integers
( Ar ) s  Ars
d1
0
(4) D  

0

0
d1k

0
0
  Dk  



 d n 
0
0 
d2 
 
0
Elementary Linear Algebra: Section 2.3, Addition
0
d 2k

0
0

0

k
 dn 


41/75

Thm 2.8： (Properties of inverse matrices)
If A is an invertible matrix, k is a positive integer, and c is a scalar
not equal to zero, then
(1) A1 is invertible and ( A1 ) 1  A
1 1
1
1 k
k
(2) Ak is invertible and ( Ak )1  
A
A

A

(
A
)

A
 

k factors
1 1
(3) cA is invertible and (cA)  A , c  0
c
1
(4) AT is invertible and ( AT )1  ( A1 )T
Elementary Linear Algebra: Section 2.3, p.67
42/75

Thm 2.9: (The inverse of a product)
If A and B are invertible matrices of size n, then AB is invertible and
( AB) 1  B 1 A1
Pf:
( AB)(B 1 A1 )  A( BB1 ) A1  A( I ) A1  ( AI ) A1  AA1  I
( B 1 A1 )( AB)  B 1 ( A1 A) B  B 1 ( I ) B  B 1 ( IB)  B 1B  I
If AB is invertible , then its inverse is unique.
So ( AB ) 1  B 1 A1

Note:
 A1 A2 A3  An 1  An1  A31 A21 A11
Elementary Linear Algebra: Section 2.3, p.68
43/75

Thm 2.10 (Cancellation properties)
If C is an invertible matrix, then the following properties hold:
(1) If AC=BC, then A=B (Right cancellation property)
(2) If CA=CB, then A=B (Left cancellation property)
Pf:
AC  BC
( AC)C 1  ( BC)C 1
(C is invertible, so C -1 exists)
A(CC 1 )  B(CC 1 )
AI  BI
A B

Note:
If C is not invertible, then cancellation is not valid.
Elementary Linear Algebra: Section 2.3, p.69
44/75

Thm 2.11: (Systems of equations with unique solutions)
If A is an invertible matrix, then the system of linear equations
Ax = b has a unique solution given by
x  A1b
Pf:
Ax  b
A1 Ax  A1b
( A is nonsingular)
Ix  A1b
x  A1b
If x1 and x2 were two solutions of equation Ax  b.
then Ax1  b  Ax2  x1  x2
(Left cancellation property)
This solution is unique.
Elementary Linear Algebra: Section 2.3, p.70
45/75

Note:
For square systems (those having the same number of equations
as variables), Theorem 2.11 can be used to determine whether the
system has a unique solution.

Note:
Ax  b
A
A
(A is an invertible matrix)

 

| b  A1 A | A1b  I | A1b
A1
| b1 | b2 |  | bn  I | A1b1 |  | A1bn 
Elementary Linear Algebra: Section 2.3, p.70
A1
46/75
Key Learning in Section 2.3
▪ Find the inverse of a matrix (if it exists).
▪ Use properties of inverse matrices.
▪ Use an inverse matrix to solve a system of linear equations.
47/75
Keywords in Section 2.3

inverse matrix: 反矩陣

invertible: 可逆

nonsingular: 非奇異

noninvertible: 不可逆

singular: 奇異

power: 冪次
48/75
2.4 Elementary Matrices

Row elementary matrix:
An nn matrix is called an elementary matrix if it can be obtained
from the identity matrix In by a single elementary operation.

Three row elementary matrices:
(1) Rij  rij ( I )
(2) Ri( k )  ri( k ) ( I )
Interchange two rows.
(k  0) Multiply a row by a nonzero constant.
(3) Rij(k )  rij(k ) (I )

Add a multiple of a row to another row.
Note:
Only do a single elementary row operation.
Elementary Linear Algebra: Section 2.4, p.74
49/75

Ex 1: (Elementary matrices and nonelementary matrices)
1 0 0
(a) 0 3 0
0 0 1
Yes (r2(3) ( I 3 ))
1 0 0
(d ) 0 0 1
0 1 0
Yes ( r23 ( I 3 ))
(b) 1 0 0
0 1 0
1 0 0
(c) 0 1 0
0 0 0
No (not square)
No (Row multiplica tion
must be by a nonzero constant)
(e) 1 0
2 1
1 0 0 
( f ) 0 2 0 
0 0 1
No (Use two elementary
row operations )
Yes (r12(2) ( I 2 ))
Elementary Linear Algebra: Section 2.4, p.74
50/75

Thm 2.12: (Representing elementary row operations)
Let E be the elementary matrix obtained by performing an
elementary row operation on Im. If that same elementary row
operation is performed on an mn matrix A, then the resulting
matrix is given by the product EA.
r(I )  E
r ( A)  EA

Notes:
(1) rij ( A)  Rij A
(2) ri( k ) ( A)  Ri( k ) A
(3) rij( k ) ( A)  Rij( k ) A
Elementary Linear Algebra: Section 2.4, p.75
51/75

Ex 2: (Elementary matrices and elementary row operation)
2
1  1  3 6
0 1 0   0
(a) 1 0 0  1  3 6  0
2
1 (r12 ( A)  R12 A)
0 0 1  3
2  1  3
2  1
1 0 0  1 0  4
1  1 0  4
1 1
1
( )
( )
 1

(b) 0
0 0 2
6  4  0 1
3  2 (r2 2 ( A)  R2 2 A)
2
0 0
 0 1
3
1 0 1
3
1
1


0  1  1
0  1
1 0 0   1
(c) 2 1 0  2  2
3  0  2
1 (r12( 2 ) ( A)  R12(5) A)
0 0 1  0
4 5 0
4 5
Elementary Linear Algebra: Section 2.4, p.75
52/75

Ex 3: (Using elementary matrices)
Find a sequence of elementary matrices that can be used to write
the matrix A in row-echelon form.
Sol:
1 3 5
0
A   1  3 0 2


2  6 2 0
0 1 0 
E1  r12 ( I 3 )  1 0 0


0 0 1
 1 0 0
1
( )
E3  r3 2 (I 3 )  0 1 0


0 0 12 
Elementary Linear Algebra: Section 2.4, p.76
 1 0 0
E2  r13( 2) ( I 3 )   0 1 0


 2 0 1
53/75
0 1 0 0 1 3 5  1  3 0 2
A1  r12 ( A)  E1 A  1 0 0 1  3 0 2  0 1 3 5
0 0 1 2  6 2 0 2  6 2 0
( 2 )
13
A2  r
 1 0 0 1  3 0 2  1  3 0 2
( A1 )  E2 A1   0 1 0 0 1 3 5  0 1 3 5
 2 0 1 2  6 2 0 0 0 2  4


1
1 0 0  1  3 0 2   1  3 0 2
( )
A3  r3 2 ( A2 )  E3 A2  0 1 0  0 1 3 5   0 1 3 5  B

1  0 0 2  4 0 0 1  2
0 0 
2

row-echelon form
1
( )
2
3
 B  E3 E2 E1 A or B  r
Elementary Linear Algebra: Section 2.4, p.76
(r13( 2) (r12 ( A)))
54/75

Row-equivalent:
Matrix B is row-equivalent to A if there exists a finite number
of elementary matrices such that
B  Ek Ek 1  E2 E1 A
Elementary Linear Algebra: Section 2.4, p.76
55/75

Thm 2.13: (Elementary matrices are invertible)
If E is an elementary matrix, then E 1 exists and
is an elementary matrix.

Notes:
(1) (Rij )1  Rij
1
( )
k
i
(2) ( Ri( k ) ) 1  R
(3) (Rij(k ) )1  Rij(k )
Elementary Linear Algebra: Section 2.4, p.77
56/75

Ex:
Elementary Matrix
0 1 0 
E1  1 0 0  R12
0 0 1
 1 0 0
E2   0 1 0  R13( 2)
 2 0 1
1
 1 0 0
( )
E3  0 1 0  R3 2
0 0 12 
Inverse Matrix
0 1 0
( R12 )  E  1 0 0  R12
0 0 1
1
(Elementary Matrix)
1 0 0
)  E  0 1 0  R13( 2) (Elementary Matrix)
2 0 1
( 2 ) 1
13
(R
1
1
1
2
1 0 0
( 2)
( R )  E  0 1 0  R3 (Elementary Matrix)
0 0 2
Elementary Linear Algebra: Section 2.4, p.77
1
( )
2 1
3
1
3
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
Thm 2.14: (A property of invertible matrices)
A square matrix A is invertible if and only if it can be written as
the product of elementary matrices.
Pf: (1) Assume that A is the product of elementary matrices.
(a) Every elementary matrix is invertible.
(b) The product of invertible matrices is invertible.
Thus A is invertible.
(2) If A is invertible, Ax  0 has only the trivial solution. (Thm. 2.11)
 A0  I 0
 Ek  E3 E2 E1 A  I
 A  E11E21E31  Ek1
Thus A can be written as the product of elementary matrices.
Elementary Linear Algebra: Section 2.4, p.77
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
Ex 4:
Find a sequence of elementary matrices whose product is
  1  2
A

3
8


Sol:
 1  2 r1( 1) 1
A



8
3
3
1
( )
1 2 r21( 2 ) 1
r2 2




0 1 
0
Therefore
2 r12( 3 ) 1 2




8
0 2 
0
I

1
1
( )
2
2
( 2 )
R21
R R12( 3) R1( 1) A  I
Elementary Linear Algebra: Section 2.4, p.78
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1
( )
2 1
2
( 2 ) 1
Thus A  ( R1( 1) ) 1 ( R12( 3) ) 1 ( R ) ( R21
)
( 2)
 R1( 1) R12(3) R2( 2 ) R21
 1 0 1 0 1 0 1 2





 0 1 3 1 0 2 0 1 

Note:
If A is invertible
Then Ek  E3 E2 E1 A  I
A1  Ek  E3 E2 E1
Ek  E3 E2 E1[ A I ]  [ I A1 ]
Elementary Linear Algebra: Section 2.4, p.78
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
Thm 2.15: (Equivalent conditions)
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 column 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.
Elementary Linear Algebra: Section 2.4, p.78
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
LU-factorization:
If the nn matrix A can be written as the product of a lower
triangular matrix L and an upper triangular matrix U, then
A=LU is an LU-factorization of A
A  LU

L is a lower triangular matrix
U is an upper triangular matrix
Note:
If a square matrix A can be row reduced to an upper triangular
matrix U using only the row operation of adding a multiple of
one row to another row below it, then it is easy to find an LUfactorization of A.
Ek  E2 E1 A  U
A  E11 E21  Ek1U
Elementary Linear Algebra: Section 2.4, p.79
A  LU
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
Ex 5: (LU-factorization)
( a ) A  1 2
1 0 
 1  3 0
(b) A  0
1 3
2  10 2
Sol: (a)
1
2
r12(-1)


A
 1 2   U
1 0
0  2
 R12( 1) A  U
 A  ( R12( 1) ) 1U  LU
( 1) 1
12
 L  (R
) R
(1)
12
1 0


1
1


Elementary Linear Algebra: Section 2.4, pp.79-80
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(b)
 1  3 0
 1  3 0
1  3 0 
( 2 )
(4)
r13
r23




A 0
1 3 
 0
1 3 0 1 3   U






2  10 2
0  4 2
0 0 14
( 4 ) ( 2 )
 R23
R13 A  U
( 4 ) 1
 A  ( R13( 2) ) 1 ( R23
) U  LU
( 4 ) 1
( 4 )
 L  ( R13( 2) ) 1 ( R23
)  R13( 2 ) R23
1 0 0 1 0 0 1 0 0
  0 1 0  0 1 0    0 1 0 
2 0 1 0  4 1 2  4 1
Elementary Linear Algebra: Section 2.4, pp.79-80
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
Solving Ax=b with an LU-factorization of A
Ax  b
If A  LU , then LUx  b
Let y  Ux, then Ly  b

Two steps:
(1) Write y = Ux and solve Ly = b for y
(2) Solve Ux = y for x
Elementary Linear Algebra: Section 2.4, p.80
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
Ex 7: (Solving a linear system using LU-factorization)
x1  3x2
 5
x2  3x3   1
2 x1  10x2  2 x3   20
Sol:
0 0  1  3 0
 1  3 0  1
A  0
1 3  0
1 0  0
1 3  LU

 


0 14
2  10 2 2  4 1 0
(1) Let y  Ux, and solve Ly  b
y1  5
 1 0 0  y1    5 
0 1 0  y2     1   y2  1
2  4 1  y3   20
y3  20  2 y1  4 y2
 20  2(5)  4(1)  14
Elementary Linear Algebra: Section 2.4, p.81
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(2) Solve the following system Ux  y
 1  3 0  x1    5 
0 1 3  x2     1 
0 0 14  x3   14
So
x3  1
x2  1  3x3  1  (3)(1)  2
x1  5  3x2  5  3(2)  1
Thus, the solution is
1
x2
 
 1
Elementary Linear Algebra: Section 2.4, p.81
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Key Learning in Section 2.4

Factor a matrix into a product of elementary matrices.

Find and use an LU-factorization of a matrix to solve a
system of linear equations.
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Keywords in Section 2.4

row elementary matrix: 列基本矩陣

row equivalent: 列等價

lower triangular matrix: 下三角矩陣

upper triangular matrix: 上三角矩陣

LU-factorization: LU分解
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Key Learning in Section 2.5

Write and use a stochastic matrix.

Use matrix multiplication to encode and decode messages.

Use matrix algebra to analyze an economic system (Leontief
input-output model).

Find the least squares regression line for a set of data.
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2.1 Linear Algebra Applied

Fight Crew Scheduling
Many real-life applications of linear systems involve
enormous numbers of equations and variables. For
example, a flight crew scheduling problem for
American Airlines required the manipulation of
matrices with 837 rows and more than 12,750,000
columns. This application of linear programming
required that the problem be partitioned into smaller
pieces and then solved on a Cray supercomputer.
Elementary Linear Algebra: Section 2.1, p.47
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2.2 Linear Algebra Applied

Information Retrieval
Information retrieval systems such as Internet search
engines make use of matrix theory and linear algebra to
keep track of, for instance, keywords that occur in a
database. To illustrate with a simplified example,
suppose you wanted to perform a search on some of the
m available keywords in a database of n documents.
You could represent the search with the m1 column
matrix x in which a 1 entry represents a keyword you
are searching and 0 represents a keyword you are not
searching. You could represent the occurrences of the m
keywords in the n documents with A, an m  n matrix
in which an entry is 1 if the keyword occurs in the
document and 0 if it does not occur in the document.
Then, the n1 matrix product ATx would represent the
number of keywords in your search that occur in each
of the n documents.
Elementary Linear Algebra: Section 2.2, p.58
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2.3 Linear Algebra Applied

Beam Deflection
Recall Hooke’s law, which states that for relatively
small deformations of an elastic object, the amount of
deflection is directly proportional to the force causing
the deformation. In a simply supported elastic beam
subjected to multiple forces, deflection d is related to
force w by the matrix equation
d = Fw
where is a flexibility matrix whose entries depend on
the material of the beam. The inverse of the flexibility
matrix, F‒1 is called the stiffness matrix. In Exercises
61 and 62, you are asked to find the stiffness matrix
F‒1 and the force matrix w for a given set of flexibility
and deflection matrices.
Elementary Linear Algebra: Section 2.3, p.64
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2.4 Linear Algebra Applied

Computational Fluid Dynamics
Computational fluid dynamics (CFD) is the computerbased analysis of such real-life phenomena as fluid
flow, heat transfer, and chemical reactions. Solving
the conservation of energy, mass, and momentum
equations involved in a CFD analysis can involve
large systems of linear equations. So, for efficiency in
computing, CFD analyses often use matrix
partitioning and -factorization in their algorithms.
Aerospace companies such as Boeing and Airbus have
used CFD analysis in aircraft design. For instance,
engineers at Boeing used CFD analysis to simulate
airflow around a virtual model of their 787 aircraft to
help produce a faster and more efficient design than
those of earlier Boeing aircraft.
Elementary Linear Algebra: Section 2.4, p.79
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2.5 Linear Algebra Applied

Data Encryption
Because of the heavy use of the Internet to conduct
business, Internet security is of the utmost importance.
If a malicious party should receive confidential
information such as passwords, personal identification
numbers, credit card numbers, social security
numbers, bank account details, or corporate secrets,
the effects can be damaging. To protect the
confidentiality and integrity of such information, the
most popular forms of Internet security use data
encryption, the process of encoding information so
that the only way to decode it, apart from a brute force
“exhaustion attack,” is to use a key. Data encryption
technology uses algorithms based on the material
presented here, but on a much more sophisticated
level, to prevent malicious parties from discovering
the key.
Elementary Linear Algebra: Section 2.5, p.87
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