Matrix Algebra: Basics and Applications in Economics

Lecturer: Dr. Monica Lambon-Quayefio, Dept. of Economics Contact Information: mplambon-quayefio@ug.edu.gh 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 1 2 3 4 1 5 8 2 0 4 2. B = 1. A = Dimensions: 4x1 Dimensions: 3x2 Number of rows is 4 and number of columns is 1 Number of rows is 3 and number of columns is 2 0 5 3 1 Dimensions: 2x4 3. C = 2 0 9 6 Number of rows is 2 and number of columns is 4

Practice Questions: Find the Dimensions 3 5 1 4 3 0 0 1 2 1 0 1 3 8 2.) 1.) 4 3.) 0 3 0 0 5 5.) 6.) 3 4.) 2

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. 0 0 d 0 1 = 0 d ( , , , ) A diag d d n d 2 = M 1 2 n n 0 0 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 * Since the matrices are equal, the corresponding elements are equal! 2x y = 1. 2x +3y 12 * Form two linear equations. 2x = y 2x + 3y = 12 * Solve the system using substitution. 2x = 3 x =3 2x = y y + 3y =12 4y =12 y = 3 2

3x + y x 2y x +3 y 2 * Write as linear equations = 2. * Combine like terms. * Solve using elimination. Example 1: Find the values for x and y using matrix equality 3x + y = x + 3 x 2y = y 2 2x + y =3 2x +6y = 4 2x + y = 3 x 3y = 2 7y = 7 y = 1 2x + 1= 3 2x = 2 x =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: a a 11 12 a a a 11 21 31 = Given a matrix then A = A' a a 21 22 a a a 12 22 32 a a 31 32 Slide 15

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 a 2 3 = If 4 c 5 is symmetric, find a, b, c? A 6 b Sol: 1 a 2 3 1 a b A = T A = = 4 c 5 A AT 2 4 c = , 3 = = , 2 5 a b c 6 b 3 5 6

Transpose: Examples 1 2 3 0 1 2 = A (a) (b) (c) = = 4 5 6 A 2 4 A 8 7 8 9 1 1 (a) Sol: 2 8 = = A T 2 A 8 2 (b) 1 3 1 4 7 = 4 5 6 A = T 2 5 8 A 7 8 9 3 6 9 (c) 0 1 0 2 1 = 2 4 A = T A 1 4 1 1 1

Properties of Transpose Matrices = T T ( ) 1 ( ) A A + ) = + T T T ( ) 2 ( A B A B = T T ( ) 3 ( ) ( ) cA c A = T T T ( ) 4 ( ) AB B A

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 + = a b c 11 11 11 a a b b c c + = a b c 11 12 11 12 11 12 where 32 31 32 31 b b a a 12 12 12 + = a a b b c c + = 21 22 21 22 21 22 a b c 21 21 21 c c 31 32 + = a b c 22 22 22 + = a b c 31 31 31 + = a b c 32 32 32 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 a a ka ka = k 11 12 11 12 a a ka ka 21 22 21 22 1 2 4 Example: Given the matrix find 3A = 3 0 1 A 2 1 2 ( ) 3 ( ) ( ) ( ) 1 3 ( ) 2 3 1 2 4 3 1 3 2 3 4 3 6 12 2 1 2 ( ) ( ) = = 3A 3 3 0 1 3 3 0 3 1 = 9 0 3 ( ) 2 3 ( ) 6 3 6 Slide 21

Practice Questions Consider the Matrices A and B 2 0 0 1 2 4 = 1 4 3 B = 3 0 1 A 1 3 2 2 1 2 Scalar Multiplication: Find the following 2A 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 Matrices- Example c c 1 4 7 1 4 30 66 11 12 = = x c c 2 5 8 2 5 36 81 21 22 c c 3 6 9 3 6 42 96 31 32 where = + + = c * * * 1 1 4 2 7 3 30 11 = + + = c * * * 1 4 4 5 7 6 66 12 = + + = c * * * 2 1 5 2 8 3 36 21 = + + = c * * * 2 4 5 5 8 6 81 22 = + + = c * * * 3 1 6 2 9 3 42 31 = + + = c * * * 3 4 6 5 9 6 96 32 Slide 24

Examples Contd. 3 5 9 2 6 2 3 1 2 . 7 4 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. x + 2y z = 1 x + 3y + 2z = 7 2x + 6y + z = 8 x y z 1 2 3 6 1 1 7 8 = 3. 1 2 1 2 Slide 25

Topic Three LAWS OF MATRIX OPERATIONS Slide 26

Properties of matrix addition and scalar multiplication: If , , , , : scalar A B C M c d m n 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. = 1 2 5 3 1 5 1 1 = = A B C 1 2 4 0 Slide 29

Trial Questions: Addition and Subtraction 4 3 2 5 2 3 1 2 0 = 3 0 1 A = 1 0 1 C = 5 4 3 B 7 1 2 4 1 2 1 3 2

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