SOAS, MSc Economics Presessional Mathematics, Statistics and Computing 14th September 2nd October 20

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SOAS, MSc EconomicsPre-sessional Mathematics,
Statistics and Computing14th September 2nd
October 2009

• Ourania Dimakou (Room 203, od1_at_soas.ac.uk)
• Satoshi Miyamura (Room 4425, sm97_at_soas.ac.uk)

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Lectures

• Mathematics 14-22 September 2009
• 1000-1300 Room V211
• Statistics 23-30 September 2009
• 1000-1300 Room V211
• Computing 18-25 September 2009
• 1400-1700 Room L62

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Sign up sheet for Computing

• Group 1
• Friday 18th and
• Wednesday 23rd September
• Group 2
• Monday 21st and
• Thursday 24th September
• Group 3
• Tuesday 22nd and
• Friday 25th September.
• Please sign up to one of the three groups on the
  sign-up sheet in front of room 203.

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Exercises

• These will be handed out daily. Although they are
  not assessed, you are strongly advised to answer
  all of the questions.

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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.

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Assessment

• One written examination of 3 hours duration
  covering both Preliminary Mathematics and
  Preliminary Statistics to be held on Friday 2nd
  October 2009.

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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.
• Abadir, K. M. and J. R. Magnus (2005) Matrix
  Algebra, Cambridge.

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Course outline

• Linear (Matrix) algebra
• Calculus.
• Exponentials and Logarithmic Functions.
• Optimization.
• Constrained Optimization.

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Induction/Welcome meeting for PG Masters
Economics

• Friday 2nd October
• V211 Vernon Sq for 2-4 pm.
• It will be a chance for students and staff to
  meet and for students to get a clearer idea about
  the options they will be taking before they
  register in the following week.
• Refreshments will be provided.

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PRELIMINARY MATHEMATICS

• LECTURE 1 MATRIX ALGEBRA

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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 10-12.
• 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.

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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/coefficien
ts
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Why study matrix algebra?
Consider the one commodity market model Matrix
algebra
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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.
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Definitions

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

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Definitions

• The members of the array is referred to as the
  elements of a matrix.

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Definitions

• As shorthand, the array in matrix can also be
  written as

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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 .

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Definitions

• In a special case where , the matrix is
  called a square matrix.
• (3 3 square matrix)

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Definitions

• A matrix composed of a single column is a column
  vector.
• (3 1 column vector)

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Definitions

• A matrix composed of a single row is a row
  vector.
• (1 3 row vector)

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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 .

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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 .
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Matrix operations

• Equality of matrices
• Also if , this implies and

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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.

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Addition and subtraction

• Example of matrices not conformable for addition
• (2 2) (2 3)

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Addition and subtraction

• Example of matrices conformable for addition
• (2 2) (2 2)

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Addition and subtraction

• Example of matrices conformable for addition
• (2 2) (2 2)

29
Scalar multiplication

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

30
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
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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
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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
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Matrix multiplication

• Example of conformable matrices

3 3
A
B
2 ? 3
3 ? 2
AB
2 ? 2
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Matrix multiplication

• Example of matrices not conformable for
  multiplication
• The product matrix AC is not defined

3 ? 2
A
C
2 ? 3
2 ? 3
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Matrix multiplication
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Matrix multiplication
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Matrix multiplication
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Matrix multiplication
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Matrix multiplication
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Matrix multiplication
or as a general expression (in this case
)
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Matrix multiplication

• Using the numerical example

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Matrix multiplication
Using the numerical example
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Matrix multiplication
Using the numerical example
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Matrix multiplication
Using the numerical example
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Matrix multiplication
Using the numerical example
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Rules of matrix operations

• 1. Matrix addition is commutative

47
Rules of matrix operations

• 1. also associative

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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.

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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.

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Rules of matrix operations

• 2. Matrix multiplication is not commutative, but
  is associative and distributive.
• Distributive law
• (pre-multiplication by )
• (post-multiplication by )

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Transpose

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

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Properties of transpose

• 1. The transpose of the transpose is the
  original matrix

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Properties of transpose

• 2. The transpose of a sum is the sum of the
  transposes.

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Properties of transpose
For example,
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Properties of transpose
For example,
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Properties of transpose

• 3. The transpose of a product is the product of
  the transposes in reverse order.

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Properties of transpose

• For example,

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Properties of transpose
For example,
59
Symmetrical matrix

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

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

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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

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Idempotent matrix

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

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Null matrix

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

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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

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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.

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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.

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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.

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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.
( )
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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.
(?)
( )
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Second order determinant

• For example,

72
Second order determinant
For example,
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Third order determinant
For a 3 3 matrix
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Diagrammatical presentation of the third order
determinant

• is expressed as sum of six product terms,
  three prefixed by plus signs and thee by minus
  signs.

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Diagrammatical presentation of the third order
determinant
is expressed as sum of six product terms,
three prefixed by plus signs and thee by minus
signs.
(?)
( )
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Diagrammatical presentation of the third order
determinant
is expressed as sum of six product terms,
three prefixed by plus signs and thee by minus
signs.
()
(?)
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Diagrammatical presentation of the third order
determinant
is expressed as sum of six product terms,
three prefixed by plus signs and thee by minus
signs.
(?)
( )
78
Diagrammatical presentation of the third order
determinant
is expressed as sum of six product terms,
three prefixed by plus signs and thee by minus
signs.
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Diagrammatical presentation of the third order
determinant (2)

• is expressed as sum of six product terms,
  three prefixed by plus signs and thee by minus
  signs.

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Diagrammatical presentation of the third order
determinant (2)
Rewrite the matrix as
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Diagrammatical presentation of the third order
determinant (2)
Rewrite the matrix as
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Diagrammatical presentation of the third order
determinant (2)
Rewrite the matrix as
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Diagrammatical presentation of the third order
determinant (2)
Rewrite the matrix as
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Diagrammatical presentation of the third order
determinant (2)
Rewrite the matrix as
85
Diagrammatical presentation of the third order
determinant (2)
Rewrite the matrix as
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Diagrammatical presentation of the third order
determinant (2)
Rewrite the matrix as
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Diagrammatical presentation of the third order
determinant (2)
Example,
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Diagrammatical presentation of the third order
determinant (2)
Example,
89
Diagrammatical presentation of the third order
determinant (2)
Example,
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Minor

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

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

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

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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,
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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,
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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.

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Third order determinant

• For example,

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Third order determinant
For example,
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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)

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Quadratic form

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

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Quadratic form
A polynomial in which each term has a uniform
degree, i.e. where the sum of exponents in each
term is uniform, 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
?
102
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.

103
Positive and negative definite matrices

• Expressing the quadratic form in matrices

104
Positive and negative definite matrices

• Expressing the quadratic form in matrices

105
Positive and negative definite matrices

• Expressing the quadratic form in matrices

106
Positive and negative definite matrices

• The quadratic form expressed in matrix

107
Positive and negative definite matrices

• A quadratic form is
• positive definite iff and
• negative definite iff and

108
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
.
109
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 .
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Positive and negative definite matrices

• Example
• Is either
  positive or negative definite?
• In matrix form
• The principal minor is
• The second principal minor is
• .

111
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.
112
Three variable quadratic form
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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 .

114
Three variable quadratic form

• A quadratic form is
• positive definite iff ,
  and
• negative definite iff ,
  and

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Three variable quadratic form

• Example

116
Three variable quadratic form
Example
117
Three variable quadratic form
Example Therefore is positive
definite.
118
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).

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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

120
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?
121
Properties of determinants

• (1) The interchange of rows and columns does not
  affect the value of a determinant, that is,

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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.
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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.
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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.
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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.
126
Properties of determinants
(6) The expansion of a determinant by alien
cofactors always yields a value of zero. (more
on this later)
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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

128
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.

129
Why is the inverse matrix useful?

• Given a system of linear equations,
• which can be written in matrix notation as
• or

130
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.

131
Linear independence

• Consider an n n matrix
• This matrix can be viewed as an ordered set of
  column vectors
• where , , etc.

132
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.

133
Linear independence

• Consider
• In this case
• If and
• we have

134
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
135
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.

136
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.

137
Rank of a matrix

• If we have
• The maximum possible rank is 2.
• ? (A) 2

138
Alien cofactors

• Recall that for

139
Alien cofactors
Recall that for The cofactor of a wrong
row or column is known as the alien cofactors.
140
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 .

141
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 .
142
Matrix inversion by the cofactor method

• An adjoint matrix is the transpose of a cofactor
  matrix.

143
Matrix inversion by the cofactor method

• Since and are conformable for
  multiplication, their product is defined

144
Matrix inversion by the cofactor method
Since and are conformable for
multiplication, their product is defined
145
Matrix inversion by the cofactor method
The principal diagonal components are the
determinant found by the Laplace expansion of
the th column
146
Matrix inversion by the cofactor method
The off-diagonal components are the determinant
expanded by th row and alien cofactor of th
row
147
Matrix inversion by the cofactor method
The off-diagonal components are the determinant
expanded by th row and alien cofactor of th
row
148
Matrix inversion by the cofactor method

• Using the rule of scalar multiplication,
• Dividing both sides of the equation by

149
Matrix inversion by the cofactor method

• Pre-multiplying by

150
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

151
Matrix inversion by the cofactor method

• Example
• Find the solution of the equation system
• Write in matrix form.
• Find the inverse of

152
Matrix inversion by the cofactor method

• Step 1.
• ( is non-singular has an
  inverse )

153
Matrix inversion by the cofactor method
Step 2. Find the cofactor matrix
154
Matrix inversion by the cofactor method
Step 3. Find the adjoint matrix
155
Matrix inversion by the cofactor method
Step 4. Obtain the inverse
156
Matrix inversion by the cofactor method
Check that the inverse is correct
157
Matrix inversion by the cofactor method
The solution can be obtained as
158
Non-homogenous equation systems

• Given a system of linear equations,
• which can be written in matrix notation as
• or

159
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.

160
Non-homogenous equation systems

• Case 1 Equations inconsistent

161
Non-homogenous equation systems
162
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.
163
Non-homogenous equation systems

• Case 2. Equations dependent

164
Non-homogenous equation systems
165
Non-homogenous equation systems
Case 2. Equations dependent A is singular
and therefore does not have an inverse. ? An
infinite set of solutions.
166
Non-homogenous equation systems

• Case 3 Consistent and independent equations

167
Non-homogenous equation systems
168
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.
169
Cramers rule
Cramers rule provides a simplified method of
solving a system of linear equations through the
use of determinants.
170
Cramers rule
Cramers 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.

171
Derivation of the Cramers rule

• Given a system of equation , the
  solution can be written as
• or

172
Derivation of the Cramers rule
173
Derivation of the Cramers rule

• We obtain the following solution values

174
Recall from yesterday

• If we have
• The Lasplace expansion by the first column

175
Recall from yesterday
If we replace the first column of A by the column
vector d The Lasplace expansion by the first
column
176
Derivation of the Cramers rule
We obtain the following solution values
177
Cramers rule

• Example
• Find the solution of the equation system
• Write in matrix form.

178
Cramers rule
179
Cramers rule

• Hence,