1. 2.1 Operations with Matrices 2.2 Properties of Matrix Operations
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
Elementary Linear Algebra 投影片設計編製者
R. Larsen et al. (6 Edition) 淡江大學 電機系 翁慶昌 教授

2. 2.1 Operations with Matrices
Matrix:
(i, j)-th entry:
row: m
column: n
size: m×n
Elementary Linear Algebra: Section 2.1, pp.46-47

3. i-th row vector row matrix j-th column vector column matrix
Square matrix: m = n
Elementary Linear Algebra: Section 2.1, p.47

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

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

6. Equal matrix: Ex 1: (Equal matrix)
Elementary Linear Algebra: Section 2.1, p.47

7. Matrix addition: Ex 2: (Matrix addition)
Elementary Linear Algebra: Section 2.1, p.47

8. 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, pp.48-49

9. Sol:
(a)
(b)
(c)
Elementary Linear Algebra: Section 2.1, p.49

10. Matrix multiplication:
Size of AB
where
Notes: (1) A+B = B+A, (2)
Elementary Linear Algebra: Section 2.1, p.50

11. Ex 4: (Find AB)
Sol:
Elementary Linear Algebra: Section 2.1, p.50

12. Matrix form of a system of linear equations:= A x b
Elementary Linear Algebra: Section 2.1, p.53

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

14. 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: 純量積
matrix multiplication: 矩陣相乘
partitioned matrix: 分割矩陣

15. 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, pp.61-62

16. 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.61

17. 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.62

18. Transpose of a matrix:
Elementary Linear Algebra: Section 2.2, p.67

19. Ex 8: (Find the transpose of the following matrix)
(b)
(c)
Sol:
(a)
(b)
(c)
Elementary Linear Algebra: Section 2.2, p.68

20. Properties of transposes:
Elementary Linear Algebra: Section 2.2, p.68

21. 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, p.68 & p.72

22. 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, p.72

23. (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

24. 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.64

25. (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.65

26. Ex 5: (An example in which cancellation is not valid) Show that AC=BC
Sol: So
But
Elementary Linear Algebra: Section 2.2, p.65

27. 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

28. 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.73

29. 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.73-74

30. 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.74-75

31. Thus
Elementary Linear Algebra: Section 2.3, p.75

32. 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.76

33. Ex 3: (Find the inverse of the following matrix)
Sol:
Elementary Linear Algebra: Section 2.3, p.76

34. So the matrix A is invertible, and its inverse is
Check:
Elementary Linear Algebra: Section 2.3, p.77

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

36. 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.79

37. 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.81

38. 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.82

39. (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.83

40. (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.83

41. Keywords in Section 2.3: inverse matrix: 反矩陣 invertible: 可逆
nonsingular: 非奇異
singular: 奇異
power: 冪次

42. 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.87

43. Ex 1: (Elementary matrices and nonelementary matrices)
Elementary Linear Algebra: Section 2.4, p.87

44. 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.89

45. Ex 2: (Elementary matrices and elementary row operation)
Elementary Linear Algebra: Section 2.4, p.88

46. 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, pp.89-90

47. row-echelon form
Elementary Linear Algebra: Section 2.4, pp.89-90

48. 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.90

49. 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.90

50. Elementary Matrix Inverse Matrix
Ex:
Elementary Matrix Inverse Matrix
(Elementary Matrix)
(Elementary Matrix)
(Elementary Matrix)
Elementary Linear Algebra: Section 2.4, p.91

51. 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.91

52. 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, pp.91-92

53. Note:
If A is invertible
Elementary Linear Algebra: Section 2.4, p.92

54. 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.93

55. 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.93

56. Ex 5: (LU-factorization)
Sol: (a)
Elementary Linear Algebra: Section 2.4, p.93

57. (b)
Elementary Linear Algebra: Section 2.4, p.94

58. 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.95

59. Ex 7: (Solving a linear system using LU-factorization)
Elementary Linear Algebra: Section 2.4, pp.95-96

60. So
Thus, the solution is
Elementary Linear Algebra: Section 2.4, pp.95-96

61. 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