1. 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:

2. 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

3. 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

4. Reading List
Sydsaeter, K. and P. Hammond, Essential Mathematics for Economic Analysis, 2nd Edition, Prentice Hall, Chapter 15
Dowling, E. T., “Introduction to Mathematical Economics”, 3rdEdition, Shaum’s Outline Series, McGraw-Hill Inc., Chapter 10
Chiang, A. C., “Fundamental Methods of Mathematical Economics”, McGraw Hill Book Co., New York, Chapter 4

5. Matrices: Basic definition and related concepts
Topic One
Matrices: Basic definition and related concepts

6. 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

7. Matrices : Dimensions Dimensions: 4x1 Dimensions: 3x2
Number of rows is 4 and number of columns is 1
Dimensions: 3x2
Number of rows is 3 and number of columns is 2
Dimensions: 2x4
Number of rows is 2 and number of columns is 4

8. Practice Questions: Find the Dimensions

9. 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.

10. Types of Matrices Row Matrix - a matrix with only one row.
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.
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.

11. Example 1: Find the values for x and y using matrix equality
* Since the matrices are equal, the corresponding elements are equal!
* Form two linear equations.
* Solve the system using substitution.

12. * Write as linear equations
* Combine like terms.
* Solve using elimination.
Example 1: Find the values for x and y using matrix equality

13. BASIC MATRIC OPERATIONS
Topic Two
BASIC MATRIC OPERATIONS

14. Matrix Operations Matrix operations involves the following
Transposition
Addition
Subtraction
Multiplication
Inversion

15. 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:
Given a matrix then

16. 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:
is symmetric, find a, b, c?
Sol:

17. Transpose: Examples
(a)
(b)
(c)
Sol:
(a)
(b)
(c)

18. Properties of Transpose Matrices

19. 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
where

20. Addition and Subtraction: Examples

21. Matrix Multiplication: Scalar
Example: Given the matrix find 3A
To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity

22. Practice Questions Consider the Matrices A and B
Scalar Multiplication: Find the following 2A
½ B
3B-A

23. 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)

24. Matrix Multiplication: 2 Matrices- Example
where

25. Examples Contd. Dimensions: 2 x 3 2 x 2
*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.

26. Laws of matrix operations
Topic Three
Laws of matrix operations

27. 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

28. 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

29. 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.

30. Trial Questions: Addition and Subtraction

31. References
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