1. SOAS, MSc Economics Preliminary Mathematics, Statistics and Computing 7th September – 25th September 2015
Sophie Van Hullen
Angelos Diamantopoulos

2. Lectures Mathematics 7-15 September 2015 10:00-13:00 Room V111
Statistics September 2015
Revision September Room V111
Computing September 2015
14:00-17:00 Room G50

3. Exercises These will be available on the Course Website.
Although they are not assessed, you are strongly advised to answer all of the questions.

4. Calculators
An electronic calculator may be helpful for parts of this course. You may use your own calculator in examinations provided that the calculator cannot store text; the make and type of calculator must be stated clearly on your examination answer book.

5. Assessment
One written examination of 3 hours duration covering both Preliminary Mathematics and Preliminary Statistics to be held on Friday 25th September 2015.

6. Textbooks
Chiang, A. C. and K. Wainwright (2005) Fundamental Methods of Mathematical Economics, Forth Edition. McGraw-Hill.
Thomas, R. L. (1999) Using Mathematics in Economics, Second Edition. Addison-Wesley.
Jacques, I. (2009) Mathematics for Economics and Business, 6th Edition, Prentice Hall.
Dowling, E. T. (2000) Introduction to Mathematical Economics, Third Edition. McGraw-Hill.

7. Course outline Linear (Matrix) algebra Calculus.
Exponentials and Logarithmic Functions.
Optimization.
Constrained Optimization.

8. PRELIMINARY MATHEMATICS
LECTURE 1 MATRIX ALGEBRA

9. Readings
Chiang, A. C. and K. Wainwright (2005) Fundamental Methods of Mathematical Economics, Forth Edition. McGraw-Hill. Chapter 4 and 5.
Dowling, E. T. (2000) Introduction to Mathematical Economics, Third Edition. McGraw-Hill. Chapters
Knowledge of elementary algebra will be assumed, including:
Functions and equations
Solutions and linear and quadratic equations
Solution of simple simultaneous equations
If you need a quick reminder of the above topics, revision booklets available from the mathcentre.ac.uk are useful, which is linked from the course website.

10. Why study matrix algebra?
Consider the one commodity market model
Three variables: quantity demanded of the commodity , the quantity supplied of the commodity , and its price Four parameters/coefficients:

11. Why study matrix algebra?
Consider the one commodity market model
Matrix algebra
(1) provides a compact way of expressing an equation system;

12. Why study matrix algebra?
Consider the one commodity market model
Matrix algebra
(2) leads to a way of testing the existence of a solution by evaluation of a determinant;
(3) provides the means of solving the equation system, if the solution exists.

13. Definitions
A matrix is a rectangular array of numbers, parameters, or variables.

14. Definitions
The members of the array are referred to as the elements of a matrix.

15. Definitions
As shorthand, the array in matrix can also be written as

16. Definitions
The number of rows and columns in a matrix define the dimension of the matrix. Since matrix contains rows and columns, it is said to be of dimension

17. Definitions
In a special case where , the matrix is called a square matrix.
(3 × 3 square matrix)

18. Definitions A matrix composed of a single column is a column vector.

19. Definitions A matrix composed of a single row is a row vector.

20. Matrix operations Equality of matrices
Two matrices and are said to be equal if and only if they have the same dimension and have identical elements in the corresponding locations in the array. In other words, if and only if for all values of and .

21. Matrix operations Equality of matrices
Two matrices and are said to be equal if and only if they have the same dimension and have identical elements in the corresponding locations in the array. In other words, if and only if for all values of and .

22. Matrix operations
Equality of matrices
Also if , this implies and

23. Addition and subtraction
Matrices are conformable for addition if and only if they have the same dimension.
Addition of and is defined as the addition of each pair of corresponding elements.

24. Addition and subtraction
Example of matrices not conformable for addition
(2 × 2) (2 × 3)
(Not conformable!)

25. Addition and subtraction
Example of matrices conformable for addition
(2 × 2) (2 × 2)

26. Addition and subtraction
Example of matrices conformable for addition
(2 × 2) (2 × 2)

27. Scalar multiplication
Multiplication of a matrix by a number (scalar) involves multiplication of every element of the matrix by the number.

28. Matrix multiplication
Conformability condition: the column dimension of the “lead” matrix must be equal to the row dimension of the “lag” matrix. A B
m  n
p  q AB

29. Matrix multiplication
In general, if is of dimension and is of dimension , the product matrix will be defined if and only if
n = p A B
m  n
p  q AB

30. Matrix multiplication
If defined, moreover, the product matrix will have the dimension the same number of rows as the lead matrix and the same number of columns as the lag matrix .
n = p A B
m  n
p  q AB
m  q

31. Matrix multiplication
Example of conformable matrices
3 = 3 A B
2  3
3  2 AB
2  2

32. Matrix multiplication
Example of matrices not conformable for multiplication
The product matrix AC is not defined
3 ≠ 2 A C
2  3
2  3

33. Matrix multiplication

34. Matrix multiplication

35. Matrix multiplication

36. Matrix multiplication

37. Matrix multiplication

38. Matrix multiplication
or as a general expression:
(in this case )

39. Matrix multiplication
Using the numerical example:

40. Matrix multiplication
Using the numerical example:

41. Matrix multiplication
Using the numerical example:

42. Matrix multiplication
Using the numerical example:

43. Matrix multiplication
Using the numerical example:

44. Rules of matrix operations
1. Matrix addition is commutative:

45. Rules of matrix operations
1. also associative

46. Rules of matrix operations
2. Matrix multiplication is not commutative, but is associative and distributive.
Not commutative:
except the scalar multiplication:
Hence the terms pre-multiply and post-multiply are often used.
See Question 6 (a) in Exercise 1 to verify this.

47. Rules of matrix operations
2. Matrix multiplication is not commutative, but is associative and distributive.
Associative law:
If conformability condition is met, any adjacent pair of matrices may be multiplied out first, provided that the product is duly inserted in the exact place of the original pair.
Use Question 9 (c) in Exercise 1 to verify this.

48. Rules of matrix operations
2. Matrix multiplication is not commutative, but is associative and distributive.
Distributive law:
(pre-multiplication by )
(post-multiplication by )

49. Transpose
When the rows and columns of a matrix are interchanged we obtain the transpose of
, which is denoted by or .

50. Properties of transpose
1. The transpose of the transpose is the original matrix

51. Properties of transpose
2. The transpose of a sum is the sum of the transposes.

52. Properties of transpose
For example,

53. Properties of transpose
For example,

54. Properties of transpose
3. The transpose of a product is the product of the transposes in reverse order.
Why reverse order?
Let be and be Then will be , and will be For equality to hold, the right-hand expression must be of the identical dimension is , as required is not even defined unless

55. Properties of transpose
For example,

56. Properties of transpose
For example,

57. Symmetrical matrix
Any matrix for which is a symmetric matrix. Symmetric matrix is a special case of square matrix.

58. Symmetrical matrix
Any matrix for which is a symmetric matrix. Symmetric matrix is a special case of square matrix.

59. Identity matrix
A square matrix with 1s in its principal diagonal and 0s in everywhere else is termed identity matrix and denoted by the symbol or , in which the subscript denotes the dimensions of the matrix
(both denoted by I)

60. Identity matrix
The identity matrix plays a role similar to that of the number 1 in scalar algebra.
(1) Multiplication of a matrix by an identity matrix leaves the original matrix unchanged.
(2) Multiplication of an identity matrix by itself leaves the identity matrix unchanged:

61. Idempotent matrix
Any matrix for which is referred to as an idempotent matrix. The identity matrix is symmetric and idempotent.

62. Null matrix
A null matrix is a matrix whose elements are all zero and can be of any dimension; it is not necessarily square.

63. Null matrix The null matrix plays the role of number 0.
(1) Addition or subtraction of the null matrix leaves the original matrix unchanged:
(2) Multiplication by a null matrix produces a null matrix.
and

64. Diagonal matrix
A diagonal matrix is one whose only nonzero entries are along the principal diagonal.
Diagonal matrix can only be idempotent only if each diagonal element is either 1 or 0. Hence identity and null matrices are special cases of a diagonal matrix.

65. Determinant
The determinant of a square matrix denoted by , is a uniquely defined scalar (number) associated with that matrix. Determinants are defined only for square matrices.

66. Second order determinant
For a 2 × 2 matrix, the determinant is obtained by multiplying the two elements in the principal diagonal and then subtracting the product of the two remaining elements.

67. Second order determinant
For a 2 × 2 matrix, the determinant is obtained by multiplying the two elements in the principal diagonal and then subtracting the product of the two remaining elements.
( + )

68. Second order determinant
For a 2 × 2 matrix, the determinant is obtained by multiplying the two elements in the principal diagonal and then subtracting the product of the two remaining elements.
(―)
( + )

69. Second order determinant
For example,

70. Second order determinant
For example,

71. Minor
The minor of denoted as is obtained by deleting the th row and th column of a given determinant.

72. Minor
The minor of denoted as is obtained by deleting the th row and th column of a given determinant.

73. Minor
The minor of denoted as is obtained by deleting the th row and th column of a given determinant.

74. Cofactor
The cofactor of denoted as is a minor with a prescribed algebraic sign attached to it.

75. Cofactor
The cofactor of denoted as is a minor with a prescribed algebraic sign attached to it.
If the sum of the two subscripts and in the minor is even, then the cofactor takes the same sign as the minor; that is,

76. Cofactor
The cofactor of denoted as is a minor with a prescribed algebraic sign attached to it.
If the sum of the two subscripts and in the minor is odd, then the cofactor takes the opposite sign as the minor; that is,

77. Laplace expansion
Using these concepts, we can express the third-order determinant as
This is known as the Laplace expansion of the third-order determinant.

78. Third order determinant
For example,

79. Third order determinant
For example,

80. n th order determinant
Laplace expansion method can be applied to evaluation of n th order determinant.
(expansion by the th row)
(expansion by the th column)

81. Properties of determinants
(1) The interchange of rows and columns does not affect the value of a determinant, that is,

82. Properties of determinants
(2) The interchange of any two rows (or any two columns) will alter the sign, but not the numerical value, of the determinant.

83. Properties of determinants
(3) The multiplication of any one row (or one column) by a scalar k will change the value of the determinant k- fold.

84. Properties of determinants
(4) The addition (subtraction) of a multiple of any row to (from) another row will leave the value of the determinant unaltered. The same holds for column.

85. Properties of determinants
(5) If one row (or column) is a multiple of another row (or column), the value of the determinant will be zero. As a special case of this, when two rows (or columns) are identical, the determinant will vanish.

86. Inverse matrix
The inverse of a matrix , denoted by , is defined only if is a square matrix, in which case the inverse is the matrix that satisfies the condition

87. Properties of inverse matrix
(1) Not every square matrix has an inverse. If the square matrix has an inverse , is said to be non- singular; if possesses no inverse, it is called a singular matrix.
(2) and are inverses of each other.
(3) If is then must also be
(4) If exists, then it is unique.

88. Why is the inverse matrix useful?
Given a system of linear equations,
which can be written in matrix notation as or

89. Why is the inverse matrix useful?
If the inverse matrix exists, the pre-multiplication of both sides of the equation by yields
Since is unique if it exists, must be a unique vector of solution values.

90. Quadratic form
A polynomial in which each term has a uniform degree, i.e. where the sum of exponents in each term is uniform (the same), is called a form. The polynomial
in which each term is of the second degree constitutes a quadratic form in two variables
and .

91. Quadratic form
A polynomial in which each term has a uniform degree, i.e. where the sum of exponents in each term is uniform (the same), is called a form. The polynomial
What restrictions must be placed upon , and when and are allowed to take any values, in order to ensure a definite sign of ?

92. Positive and negative definiteness
A quadratic form is said to be
• positive definite if
• positive semi-definite if
• negative semi-definite if
• negative definite if
regardless of the values of the variables in the quadratic form, not all zero.

93. Positive and negative definite matrices
Expressing the quadratic form in matrices

94. Positive and negative definite matrices
Expressing the quadratic form in matrices

95. Positive and negative definite matrices
Expressing the quadratic form in matrices

96. Positive and negative definite matrices
The quadratic form expressed in matrix
The determinant of the 2 × 2 coefficient
matrix is referred to as the
discriminant of the quadratic form , and denoted by

97. Positive and negative definite matrices
A quadratic form is
positive definite iff and
negative definite iff and

98. Positive and negative definite matrices
A quadratic form is
positive definite iff and
negative definite iff and
is a sub-determinant of that consists of the first element on the principal diagonal, called the first principal minor of

99. Positive and negative definite matrices
A quadratic form is
positive definite iff and
negative definite iff and
Similarly since involves the first and second
elements on the principal diagonal, it is called the second principal minor of

100. Positive and negative definite matrices
Example
Is either positive or negative definite?
In matrix form:
The principal minor is
The second principal minor is .

101. Positive and negative definite matrices
Example
Is either positive or negative definite?
In matrix form:
The principal minor is
The second principal minor is
Therefore is positive definite.

102. Three variable quadratic form

103. Three variable quadratic form
A total of three principal minors can be found from the discriminant:
where denotes the th principal minor of the discriminant .

104. Three variable quadratic form
A quadratic form is
• positive definite iff , and
• negative definite iff , and

105. Three variable quadratic form
Example

106. Three variable quadratic form
Example

107. Three variable quadratic form
Example
Therefore is positive definite.

108. n variable quadratic forms
In general for
the quadratic form is
• positive definite iff …
• negative definite iff …
(all odd-numbered principal minors are negative and all even- numbered ones are positive).

109. Why are we interested in the sign of quadratic forms?
At this point we have discussed quadratic forms as one application of determinants.
Later in the course we will use this to consider problems such as follows:

110. Why are we interested in the sign of quadratic forms?
At this point we have discussed quadratic forms as one application of determinants.
Later in the course we will use this to consider problems such as follows:
Let us consider a firm that is a price taker, produces and sells two goods, goods 1 and goods 2 in perfectly competitive markets, and has a profit function:
Can we ensure that at the optimal level of output for goods 1 and 2 we have maximum profit?

111. Linear independence Consider an n × n matrix
This matrix can be viewed as an ordered set of column vectors:
where , , etc.

112. Linear independence Definition
A set of vectors is linearly dependent, if and only if we can find a set of scalars, (not all of which are zero) such that
If such a set of scalars cannot be found, the vectors are linearly independent.

113. Linear independence
Consider
In this case
If and
we have

114. Non-singular matrix and linear independence
For the non-singularity of a matrix, its rows (or columns) must be linearly independent, i.e. none must be a linear combination of the rest.
The condition for inverse matrix corresponds to the condition for simultaneous equations to have a unique solution:
Condition for simultaneous equations
Condition for inverse matrix
Same number of equations as unknowns
Square matrix (same number of rows and columns)
Equations must be consistent and functionally linearly independent
Non-singular matrix

115. Test of non-singularity
From property (5) of the determinant, if the rows (column) of a matrix is dependent Hence If
→ there is row (column) independence in matrix
→ is non-singular
→ exists
→ a unique solution exists.

116. Rank of a matrix
The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix.
An non-singular matrix is of rank . In general, the rank of an matrix can be at most or , whichever is smaller.
Alternatively we can define the rank of an matrix as the maximum number of a non-vanishing determinant that can be constructed from the rows and columns of that matrix.

117. Rank of a matrix
If we have
The maximum possible rank is 2.
ρ (A) = 2

118. Alien cofactors
Recall that for

119. Alien cofactors Recall that for
The cofactor of a “wrong” row or column is known as the alien cofactors.

120. Matrix inversion by the cofactor method
Given a non-singular matrix
A cofactor matrix, denoted by , is given as a matrix in which every element is replaced with its cofactor

121. Matrix inversion by the cofactor method
Given a non-singular matrix
A cofactor matrix, denoted by , is given as a matrix in which every element is replaced with its cofactor

122. Matrix inversion by the cofactor method
An adjoint matrix is the transpose of a cofactor matrix.

123. Matrix inversion by the cofactor method
Since and are conformable for multiplication, their product is defined

124. Matrix inversion by the cofactor method
Since and are conformable for multiplication, their product is defined

125. Matrix inversion by the cofactor method
The principal diagonal components are the determinant found by the Laplace expansion of the
th column

126. Matrix inversion by the cofactor method
The off-diagonal components are the determinant expanded by th row and alien cofactor of th row

127. Matrix inversion by the cofactor method
The off-diagonal components are the determinant expanded by th row and alien cofactor of th row

128. Matrix inversion by the cofactor method
Using the rule of scalar multiplication,
Dividing both sides of the equation by

129. Matrix inversion by the cofactor method
Pre-multiplying by

130. Matrix inversion by the cofactor method
To recap the cofactor method:
Step 1. Find and check that ;
Step 2. Find the cofactor matrix
Step 3. Find the adjoint matrix ;
Step 4. Obtain the desired inverse

131. Matrix inversion by the cofactor method
Example
Find the solution of the equation system
Write in matrix form.
Find the inverse of

132. Matrix inversion by the cofactor method
Step 1.
( is non-singular; has an inverse )

133. Matrix inversion by the cofactor method
Step 2. Find the cofactor matrix

134. Matrix inversion by the cofactor method
Step 3. Find the adjoint matrix

135. Matrix inversion by the cofactor method
Step 4. Obtain the inverse

136. Matrix inversion by the cofactor method
Check that the inverse is correct:

137. Matrix inversion by the cofactor method
The solution can be obtained as:

138. Non-homogenous equation systems
Given a system of linear equations,
which can be written in matrix notation as or

139. Non-homogenous equation systems
In the equation system, if d = 0 the equation system will be Ax = O, where O is a zero vector. This special case is referred to as a homogenous-equation system.
In contrast if d ≠ 0 we have a non-homogenous equation system.

140. Non-homogenous equation systems
Case 1: Equations inconsistent

141. Non-homogenous equation systems

142. Non-homogenous equation systems
Case 1: Equations inconsistent
Writing the system in matrix form
A is singular and therefore does not have an inverse. → No solution.

143. Non-homogenous equation systems
Case 2. Equations dependent

144. Non-homogenous equation systems

145. Non-homogenous equation systems
Case 2. Equations dependent
A is singular and therefore does not have an inverse. → An infinite set of solutions.

146. Non-homogenous equation systems
Case 3: Consistent and independent equations

147. Non-homogenous equation systems

148. Non-homogenous equation systems
Case 3: Consistent and independent equations
A is non-singular and therefore has an inverse.
A unique vector of solutions can be found.

149. Cramer’s rule
Cramer’s rule provides a simplified method of solving a system of linear equations through the use of determinants.

150. Cramer’s rule
Cramer’s rule provides a simplified method of solving a system of linear equations through the use of determinants.
where is the th unknown variable in the system of equations, and is the determinant of a special matrix formed from the original coefficient matrix by replacing the th column with the column vector of constants.

151. Derivation of the Cramer’s rule
Given a system of equation , the solution can be written as or

152. Derivation of the Cramer’s rule

153. Derivation of the Cramer’s rule
We obtain the following solution values

154. Recall from yesterday…
If we have
The Lasplace expansion by the first column

155. Recall from yesterday…
If we replace the first column of A by the column vector d…
The Lasplace expansion by the first column

156. Application of Cramer’s rule
We obtain the following solution values

157. Cramer’s rule Example Find the solution of the equation system
Write in matrix form.

158. Cramer’s rule

159. Cramer’s rule
Hence,