Ch3-reference - 43 Chapter 4 - Matrices The theory of...

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43 Chapter 4 -- Matrices The theory of matrices, discovered by Arthur Cayley in the 19th century and advanced by others, was initially regarded as a mathematical curiosity without much practical application. However, since the Second World War, developments in matrix algebra have allowed it to become a powerful tool for summarizing and analysing systems of equations. In this introduction to the topic, we hope to show some of its utility, however much of its power must be left to higher level courses. 1. Introduction A matrix is a rectangular array of numbers, generally either one dimension or two dimensions, e.g., 5 A = 3 -1 5 b = [1 3 4 1 2] c = 2 4 2 9 1 We can represent student enrolments by a matrix Year First Year Second Year Third Year 1993 700 85 35 1994 750 90 30 1995 1050 120 45 1996 1200 150 60 We could represent this information in a matrix, say, X = 700 85 35 750 90 30 1050 120 45 1200 150 60 A matrix is generally denoted by a letter in bold type. We will follow the convention which denotes a two dimensional matrix by a capital letter (e.g., A ) and a one with only a single row or column (called a vector) by a lower case letter (e.g., b , c ). A matrix is defined not only by its elements, but by its dimensions, i. e., the number of rows and columns it possesses. The matrix A above is said to be of order 2 x 3, as it has two rows and three columns. (Sometimes the term dimension is used interchangeably with the term order . Thus A could be called a two dimensional matrix, and it would be said to have dimension 2x3.) Similarly b is said to be a matrix of order 1 x 5 (or equivalently, a row vector of order 5). The location within a matrix of an element, a ij , is denoted by its row (i) and column (j) subscripts. The first subscript is that of the row, and the second subscript is that of the column. Thus for the matrix defined by A above the element a 21 is the value 4, as it is in the second row, first column the element a 13 is the value 5 as it is in the first row, second column.
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44 Two matrices are said to be equal if they are of identical dimensions, with the same elements in exactly the same positions. Thus for example the matrix A above is equal to the matrix B below, but not to the matrix C. B = 3 -1 5 C = 3 -1 9 4 2 9 4 2 5 Matrix C has the same elements as A , but as they are in a different order, A C . 2. Matrix Addition Consider three students' results in their semester examinations in Accounting and Econometrics. These could be represented by two 3 x 2 matrices. First Semester Second Semester Accounting Econometrics Accounting Econometrics Student 1 45 65 55 73 Student 2 58 60 67 72 Student 3 75 70 80 78 We can add both semesters Accounting results for Student 1 (45 + 55 = 100), and similarly for each of the other students. If our results are summarized by two matrices, (where S 1 are first semester results, S 2 for second semester), the elements of their sum represent totals for the year, T . The results follow S 1 = 45 65 58 60 75 70 S 2 = 55 73 67 72 80 78
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This note was uploaded on 05/27/2011 for the course WORK 2218 taught by Professor - during the One '09 term at University of Sydney.

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Ch3-reference - 43 Chapter 4 - Matrices The theory of...

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