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ECON 213
Elements of Mathematics for
Economists
Session 4: Introduction to Matrix Algebra- Part One
Lecturer: Dr. Monica Lambon-Quayefio, Dept. of Economics
Contact Information: [email protected]
College of Education
School of Continuing and Distance Education
2014/2015 – 2016/2017
Session Overview
• Matrix algebra has several uses in economics as well as other fields
of study. One important application of Matrices is that it enables us
to handle a large system of equations. It also allows us to test for
the existence of a solution to a system of equations even before we
attempt solving them. This session explains the basic concepts and
terminologies used in matrices on which the subsequent sessions
on matrices will build on.
• Objectives:
– Describe what matrices are and understand the different terminology
used in matrices
– Know the different type of matrices
– Be able to express a system of linear equations in matrix form
– Know the basic matrix operations such as addition, subtraction and
multiplication
– Understand and prove the commutative, associative and distributive laws
of matrices
Slide 2
Session Outline
The key topics to be covered in the session are as follows:
• Matrices: Basic definition, related concepts and types
• Basic Matrix Operations
• Laws of Matrix Operations
Slide 3
Reading List
• Sydsaeter, K. and P. Hammond, Essential Mathematics for
Economic Analysis, 2nd Edition, Prentice Hall, 2006- Chapter 15
• Dowling, E. T., “Introduction to Mathematical Economics”,
3rdEdition, Shaum’s Outline Series, McGraw-Hill Inc., 2001.Chapter 10
• Chiang, A. C., “Fundamental Methods of Mathematical
Economics”, McGraw Hill Book Co., New York, 1984.- Chapter
4
Slide 4
Topic One
MATRICES: BASIC DEFINITION AND
RELATED CONCEPTS
Slide 5
Matrices
• Matrix - a rectangular array of variables or constants in
horizontal rows and vertical columns enclosed in brackets.
• Element - each value in a matrix; either a number or a
constant.
• Dimension - number of rows by number of columns of a
matrix.
• **A matrix is named by its dimensions
Slide 6
Matrices : Dimensions
2
1. A = 
0

4
1
5 

8 

2.
B
1 


2 
=  

3 
 
4 

Dimensions: 3x2
Dimensions: 4x1
Number of rows is 3 and
number of columns is 2
Number of rows is 4 and
number of columns is 1
 0 5 3 1
3. C = 


2
0
9
6


Dimensions: 2x4
Number of rows is 2 and
number of columns is 4
Practice Questions: Find the Dimensions
 3 5 


1

1.)  4
4 

 
0


4.) 
2 
 3 0
2.) 

0
3


 5
5.)  
 
1 2 3
3.) 0 1 8
0 0 1
6.)  3
Leading Diagonal and Trace
• The leading/principal diagonal of a matrix is the elements
of the diagonal that runs from top left top corner of the
matrix to the bottom right as circled in the matrix below.
• When all other elements of a matrix are zero except the
elements of the leading diagonal, the matrix is referred
to as a diagonal matrix.
• The trace of a matrix is the sum of all the elements of the
leading diagonal.
•
d1 0

A  diag (d1 , d 2 ,, d n )   0 d 2


0 0

 0
 0
  M nn
 
 d n 
Slide 9
Types of Matrices
•
Column Matrix - a matrix with only one column.
• Row Matrix - a matrix with only one row.
• Square Matrix - a matrix that has the same number of rows and columns.
• Identity Matrix - An identity matrix is a square which has 1 for every element on the
principal diagonal from left to right and 0 everywhere else.
1
0

0

0
0
1
0
0
0
0
1
0
0
0
0

1
• Equal Matrices - two matrices that have the same dimensions and each element of one
matrix is equal to the corresponding element of the other matrix.
• *The definition of equal matrices can be used to find values when elements of the
matrices are algebraic expressions.
Example 1: Find the values for x and y using
matrix equality
 2x  y 
1. 
  

2x  3y  12 
2x  y
2x  3y  12
* Since the matrices are equal, the
corresponding elements are equal!
* Form two linear equations.
* Solve the system
using substitution.
2x  y
4y  12
y  3y  12
y3
2x  3
3
x
2
3x  y  x  3
2. 
 


x

2y
y

2

 

* Write as linear equations
* Combine like terms.
* Solve using elimination.
Example 1: Find the values for x and y using
matrix equality
3x  y  x  3
2x  y  3
2x  y  3
x  2y  y  2
x  3y  2
2x  6y  4
2x  1  3
2x  2
x 1
7y  7
y1
Topic Two
BASIC MATRIC OPERATIONS
Slide 13
Matrix Operations
•
•
•
•
•
•
Matrix operations involves the following
Transposition
Addition
Subtraction
Multiplication
Inversion
Slide 14
Matrix Operations: Transpose
• The transpose of a matrix is a new matrix that is formed by
interchanging the rows and columns.
• The transpose of A is denoted by A' or (AT)
• Examples:
•
 a11
Given a matrix A  a 21
a31
a12 
a 22 
a32 
then A'
 a11
 
a12
Slide 15
a 21
a 22
a31 
a32 
Matrix Operations: Transpose
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

b
Sol:
1
A  a

b
4
c
2
4
c

5

6
is symmetric, find a, b, c?
3
1
5 AT  2


6
3
a
4
5
b
c

6
A  AT
 a  2, b  3, c  5
Transpose: Examples
 2
(a) A   
8 
Sol: (a)
 2
A 
8 
(b)
 1 2 3
A   4 5 6


7 8 9
 AT  2 8
 1 2 3
1
A  4 5 6  AT  2



7 8 9
3
(c)
1
0
0
T
A  2
4
A 


1
 1  1
(b)
4 7
5 8

6 9
2 1
4  1
1
0
(c) A  2 4


 1  1
Properties of Transpose Matrices
(1) ( AT )T  A
(2) ( A  B)T  AT  BT
(3) (cA) T  c( AT )
(4) ( AB)T  BT AT
Matrices: Addition and Subtraction
• Two matrices may be added (or subtracted) if and
only if they are the same order or dimension.
• Simply add (or subtract) the corresponding elements.
So, A + B = C yields
•
 a11
a
 21
a31
a12  b11
a 22   b21
a32  b31
b12   c11
b22   c 21
b32  c31
c12 
c 22 
c32 
a11  b11  c11
where
a12  b12  c12
a21  b21  c 21
a22  b22  c 22
a31  b31  c31
a32  b32  c32
Slide 19
Addition and Subtraction: Examples
 1 2  1 3  1  1 2  3  0 5
 0 1   1 2   0  1 1  2   1 3

 
 
 

 1  1  1  1 
0 
  3   3    3  3   0
•     
 

 2  2  2  2
0
Slide 20
Matrix Multiplication: Scalar
• To multiply a scalar times a matrix, simply multiply
each element of the matrix by the scalar quantity
 a11 a12   ka11 ka12 
k


a
a
 21 22  ka21 ka22 
• Example: Given the
•
 1 2 4
matrix A   3 0  1
 2 1 2
find 3A
 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


 


 2 1 2  32 31 32
6
 6 3
Slide 21
Practice Questions
• Consider the Matrices A and B
 1 2 4
A   3 0  1
 2 1 2
0 0
 2
B   1  4 3


3 2
 1
• Scalar Multiplication: Find the following
• 2A
•½ B
• 3B-A
Matrix Multiplication: 2 Matrices
• To multiply a matrix times a matrix, we write
AB (A times B)
• In order to multiply matrices, they must be
CONFORMABLE meaning that the number of
columns in A must equal the number of rows in B
• So we have :
A  B = C
(m  n)  (n  p) = (m  p)
– if (m  n)  (p  n) = cannot be done
– (1  n)  (n  1) = a scalar (1x1)
Slide 23
Matrix Multiplication: 2 MatricesExample
1 4 7 1 4 c11 c12  30 66
2 5 8 x 2 5  c
  36 81
c

 
  21 22  

3 6 9 3 6 c31 c32  42 96
• where
c11  1 * 1  4 * 2  7 * 3  30
c12  1 * 4  4 * 5  7 * 6  66
c 21  2 * 1  5 * 2  8 * 3  36
c 22  2 * 4  5 * 5  8 * 6  81
c31  3 * 1  6 * 2  9 * 3  42
c32  3 * 4  6 * 5  9 * 6  96
Slide 24
Examples Contd.
3 9 2  2 1
2. 
 3 4 
5
7
6



Dimensions: 2 x 3
2x2
*The number of columns in the first matrix does not match
the number of rows in the second matrix so the two
matrices cannot be multiplied.
1 2 1  x  1 
3. 1 3 2   y   7 
 2 6 1   z  8 
x  2y  z  1
x  3y  2z  7
2x  6y  z  8
Slide 25
Topic Three
LAWS OF MATRIX OPERATIONS
Slide 26
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
Commutative, Associative and
Distributive Laws
• Commutative: A + B = B + A
• Associative: (A+B)+C = A(B+C)
(AB)C= A(BC)
• Distributive: A(=B+C)= AB+AC
(A+B)+C = AC + AB
Slide 28
Practice: Proofs
Show whether or not the following relations are true given the
following matrices A, B and C.
• A (B+C) = AB +AC
• (AB)C=A(BC)
• Given the following matrices.
 5  3
A

2

1


 1 5
B

4
0


 1 1
C

1
2


Slide 29
Trial Questions: Addition and
Subtraction
 4 3 2
A   3 0  1
 7 1 2
 1  2 0
B   5  4 3
 1
3 2
 5 2  3
C   1 0  1
 4 1
2
References
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Slide 31