1. CHAPTER 2 MATRICES 2.1 Operations with Matrices
2.2 Properties of Matrix Operations
2.3 The Inverse of a Matrix
2.4 Elementary Matrices
2.5 Applications of Matrix Operations
Elementary Linear Algebra 投影片設計編製者
R. Larson (7 Edition) 淡江大學 電機系 翁慶昌 教授

2. 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)

3. 2.1 Operations with Matrices
Matrix:
(i, j)-th entry:
row: m
column: n
size: m×n
Elementary Linear Algebra: Section 2.1, p.40

4. i-th row vector row matrix j-th column vector column matrix
Square matrix: m = n
Elementary Linear Algebra: Section 2.1, p.40

5. Diagonal matrix: Trace:
Elementary Linear Algebra: Section 2.1, Addition

6. Ex:
Elementary Linear Algebra: Section 2.1, Addition

7. Equal matrix: Ex 1: (Equal matrix)
Elementary Linear Algebra: Section 2.1, p.40

8. Matrix addition: Ex 2: (Matrix addition)
Elementary Linear Algebra: Section 2.1, p.41

9. Scalar multiplication:
Matrix subtraction:
Ex 3: (Scalar multiplication and matrix subtraction)
Find (a) 3A, (b) –B, (c) 3A – B
Elementary Linear Algebra: Section 2.1, p.41

10. Sol:
(a)
(b)
(c)
Elementary Linear Algebra: Section 2.1, p.41

11. Matrix multiplication:
Size of AB
where
Notes: (1) A+B = B+A, (2)
Elementary Linear Algebra: Section 2.1, p.42 & p.44

12. Ex 4: (Find AB)
Sol:
Elementary Linear Algebra: Section 2.1, p.43

13. Matrix form of a system of linear equations:= A x b
Elementary Linear Algebra: Section 2.1, p.45

14. Partitioned matrices: submatrix
Elementary Linear Algebra: Section 2.1, Addition

15. Linear combination of column vectors:
英文版未出現 15
Elementary Linear Algebra: Section 2.1, p.46

16. Ex 7： (Solve a system of linear equations)
(infinitely many solutions)
英文版未出現 16
Elementary Linear Algebra: Section 2.1, p.47

17. 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..

18. 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

19. 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:
Elementary Linear Algebra: Section 2.2, 52-55

20. Properties of matrix addition and scalar multiplication:
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

21. Properties of zero matrices:
Notes:
0m×n: the additive identity for the set of all m×n matrices
–A: the additive inverse of A
Elementary Linear Algebra: Section 2.2, p.53

22. 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)
英文版未出現
Properties of identity matrix: 22
Elementary Linear Algebra: Section 2.2, p.54 & p.56

23. Transpose of a matrix:
Elementary Linear Algebra: Section 2.2, p.57

24. Ex 8: (Find the transpose of the following matrix)
(b)
(c)
Sol:
(a)
(b)
(c)
Elementary Linear Algebra: Section 2.2, p.57

25. Properties of transposes:
Elementary Linear Algebra: Section 2.2, p.57

26. A square matrix A is symmetric if A = AT
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:
is symmetric, find a, b, c?
Sol:
Elementary Linear Algebra: Section 2.2, Addition

27. is a skew-symmetric, find a, b, c?
Ex:
is a skew-symmetric, find a, b, c?
Sol:
Note:
is symmetric
Pf:
Elementary Linear Algebra: Section 2.2, Addition

28. (Commutative law for multiplication)
Real number:
ab = ba
(Commutative law for multiplication)
Matrix:
Three situations:
(Sizes are not the same)
(Sizes are the same, but matrices are not equal)
Elementary Linear Algebra: Section 2.2, Addition

29. Sow that AB and BA are not equal for the matrices.
Ex 4:
Sow that AB and BA are not equal for the matrices.
and
Sol:
Note:
Elementary Linear Algebra: Section 2.2, p.55

30. (1) If C is invertible, then A = B
Real number:
(Cancellation law)
Matrix:
(1) If C is invertible, then A = B
(Cancellation is not valid)
Elementary Linear Algebra: Section 2.2, p.55

31. Ex 5: (An example in which cancellation is not valid) Show that AC=BC
Sol: So
But
Elementary Linear Algebra: Section 2.2, p.55

32. 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.

33. 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

34. 2.3 The Inverse of a Matrix Inverse matrix: Consider
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

35. 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:
Consequently, the inverse of a matrix is unique.
Notes:
(1) The inverse of A is denoted by
Elementary Linear Algebra: Section 2.3, pp.62-63

36. Find the inverse of a matrix by Gauss-Jordan Elimination:
Ex 2: (Find the inverse of the matrix)
Sol:
Elementary Linear Algebra: Section 2.3, pp.63-64

37. Thus
Elementary Linear Algebra: Section 2.3, pp.63-64

38. If A can’t be row reduced to I, then A is singular.
Note:
If A can’t be row reduced to I, then A is singular.
Elementary Linear Algebra: Section 2.3, p.64

39. Ex 3: (Find the inverse of the following matrix)
Sol:
Elementary Linear Algebra: Section 2.3, p.65

40. So the matrix A is invertible, and its inverse is
Check:
Elementary Linear Algebra: Section 2.3, p.65

41. Power of a square matrix:
Elementary Linear Algebra: Section 2.3, Addition

42. 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
Elementary Linear Algebra: Section 2.3, p.67

43. Thm 2.9: (The inverse of a product)
If A and B are invertible matrices of size n, then AB is invertible and
Pf:
Note:
Elementary Linear Algebra: Section 2.3, p.68

44. 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:
Note:
If C is not invertible, then cancellation is not valid.
Elementary Linear Algebra: Section 2.3, p.69

45. (Left cancellation property)
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
Pf:
( A is nonsingular)
(Left cancellation property)
This solution is unique.
Elementary Linear Algebra: Section 2.3, p.70

46. (A is an invertible matrix)
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:
(A is an invertible matrix)
Elementary Linear Algebra: Section 2.3, p.70

47. 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.

48. 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

49. 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:
Interchange two rows.
Multiply a row by a nonzero constant.
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

50. Ex 1: (Elementary matrices and nonelementary matrices)
Elementary Linear Algebra: Section 2.4, p.74

51. 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.
Notes:
Elementary Linear Algebra: Section 2.4, p.75

52. Ex 2: (Elementary matrices and elementary row operation)
Elementary Linear Algebra: Section 2.4, p.75

53. 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:
Elementary Linear Algebra: Section 2.4, p.76

54. row-echelon form
Elementary Linear Algebra: Section 2.4, p.76

55. Row-equivalent:
Matrix B is row-equivalent to A if there exists a finite number of elementary matrices such that
Elementary Linear Algebra: Section 2.4, p.76

56. Thm 2.13: (Elementary matrices are invertible)
If E is an elementary matrix, then exists and
is an elementary matrix.
Notes:
Elementary Linear Algebra: Section 2.4, p.77

57. Elementary Matrix Inverse Matrix
Ex:
Elementary Matrix Inverse Matrix
(Elementary Matrix)
(Elementary Matrix)
(Elementary Matrix)
Elementary Linear Algebra: Section 2.4, p.77

58. Thus A can be written as the product of elementary matrices.
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.
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.
Pf:
(2) If A is invertible, has only the trivial solution. (Thm. 2.11)
Thus A can be written as the product of elementary matrices.
Elementary Linear Algebra: Section 2.4, p.77

59. Find a sequence of elementary matrices whose product is
Ex 4:
Find a sequence of elementary matrices whose product is
Sol:
Elementary Linear Algebra: Section 2.4, p.78

60. Note:
If A is invertible
Elementary Linear Algebra: Section 2.4, p.78

61. 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

62. If the nn matrix A can be written as the product of a lower
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
L is a lower triangular matrix
Note:
U is an upper triangular matrix
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 LU-factorization of A.
Elementary Linear Algebra: Section 2.4, p.79

63. Ex 5: (LU-factorization)
Sol: (a)
Elementary Linear Algebra: Section 2.4, pp.79-80

64. (b)
Elementary Linear Algebra: Section 2.4, pp.79-80

65. Solving Ax=b with an LU-factorization of A
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

66. Ex 7: (Solving a linear system using LU-factorization)
Elementary Linear Algebra: Section 2.4, p.81

67. So
Thus, the solution is
Elementary Linear Algebra: Section 2.4, p.81

68. 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.

69. 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

70. 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.

71. 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

72. 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

73. 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

74. 2.4 Linear Algebra Applied
Computational Fluid Dynamics
Computational fluid dynamics (CFD) is the computer-based 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

75. 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