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Transcript
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 nn 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 y3 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 y1 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 31 32 34 3 6 12 3A 3 3 0 1 3 3 30 3 1 9 0 3 2 1 2 32 31 32 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 mn , 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 • XXXXXXXXXXXXXXXXXXXXXXXXXX Slide 31