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# Chapter 3 - &Econometrics...

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ECMT1020: Chapter 3 1 Discipline of Operations Management & Econometrics ECMT1020: Business & Economic Statistics B Dr Boris Choy ECMT 1020 Chapter 3 Matrix Algebra

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ECMT1020: Chapter 3 2 Topics covered 1. What is a matrix? 2. Matrix addition, subtraction & multiplication 3. Transpose, determinant & inverse of a matrix 4. Applications References ‐ Woodridge, J.M. Introductory Econometrics . Thomason. (Appendix D) ‐ Wackerly, D.D., Mendenhall, W. & Scheaffer, R.L. Mathematical Statistics with Applications. Thomson. (Appendix 1) ‐ Griffiths W.E., Hill, R.C. & Judge, G.G. Learning and Practicing Econometrics . Wiley. (Ch.3) ‐ Gujarati, D.N. & Porter, D.C. Basic Econometrics. McGraw Hill. (Appendix B) ‐ Any mathematical texts on matrix algebra
ECMT1020: Chapter 3 3 Learning Objectives Able to perform basic matrix operations. Able to solve system of equations using matrix algebra. Understand the application of matrix algebra in regression analysis

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ECMT1020: Chapter 3 4 The Matrix Download a picture from http://www.imdb.com/media/rm1818663168/tt0234215
ECMT1020: Chapter 3 5 What is a Matrix? A matrix is a rectangle array of numbers An m x n matrix has m rows and n columns (of order m x n) An 1 x n matrix is called a row vector An n x 1 matrix is called a column vector A matrix is usually denoted by a uppercase boldface letter A vector is usually denoted by a lowercase boldface letter An m x n matrix is given by where a ij , i=1 ,..., m ; j=1 ,..., n is the ( i , j )‐th element of matrix A . m n m m n n ij a a a a a a a a a a 2 1 2 22 21 1 12 11 A

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ECMT1020: Chapter 3 6 Examples A 2x3 matrix A row vector of length 4 A column vector of length 2 7 2 4 5 2 3 9 2 4 5 1 3 y x A
ECMT1020: Chapter 3 7 A real Application Student enrolment to a first year business statistics unit The data can be presented by a 4 x 3 matrix X Year Year 1 students Year 2 students Year 3 students 2006 726 85 35 2007 951 91 29 2008 1050 125 50 2009 1209 145 68 68 145 1209 50 125 1050 29 91 951 35 85 726 X

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ECMT1020: Chapter 3 8 More Matrices Zero matrix , denoted by 0 , is an m x n matrix whose elements are all zero Square matrix is an m x m matrix Symmetric matrix is a square matrix where a ij = a ji for all i , j . Note: i = row index, j = column index Diagonal matrix is a square matrix whose off‐diagonal elements are zero, i.e. a ij = 0 for all i j Identity matrix , denoted by I , is a diagonal matrix whose diagonal elements are one and off‐diagonal elements are zero, i.e. a ii = 1 and a ij = 0 for all i j Lower (Upper) triangular matrix is a square matrix whose elements in the triangle above (below) the diagonal are zero.
ECMT1020: Chapter 3 9 More Matrices Identify these matrices

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