Summary of Matrix Algebra - Chapter 4 1 Introduction...

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Chapter 4 -- Matrices 1. Introduction There are a certain number of terms that are used in matrix algebra with which you need to become familiar. Some of these terms are:- array, dimension, order, element, scalar, vector, identity matrix, transpose, inverse, determinant, reciprocal, adjoint, cofactor. A matrix is a rectangular array of numbers, generally either one dimension or two dimensions and is usually denoted by a letter in bold type, e.g., A = 3 1 5 4 2 9 - b = [ ] 1 3 4 1 2 c = 5 2 1 Note that we have denoted a two-dimensional matrix with a capital letter, e.g. A , and a one-dimensional matrix with a lower-case letter, e.g. b. One way of defining a matrix is by its dimensions. Here the matrix, A , is said to be of order 2 x 3, as it has 2 rows and 3 columns. Note we always state the number of rows first, then the number of columns. The matrix, b , is usually referred to as a vector since b has only 1 row (its dimensions are 1 x 5 as it has 1 row and 5 columns). Similarly, c is also a vector as it only has 1 column and it is of order 3 x 1 as it has 3 rows and 1 column. Example 1. In recent years, the student enrolments in Econometrics in first , second and third year have been as follows: 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. X = 700 85 35 750 90 30 1050 120 45 1200 150 60 A matrix is also defined by its individual elements. We denote the location (or position) of an element in the matrix by specifying the row and column in which the element is found. For example, in the matrix X above, the number 750 is in the 2 nd row and the 1 st column. We would denote this element as a 21 . Note the row number is first and the column number is second. In general terms, the a ij element is in row i and column j. Another example is that the number 35 would be represented by a 13 as it is in the 1 st row and the 3 rd column. When are two matrices equal?
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Two matrices are equal if they have identical dimensions and the same elements in exactly the same positions. Thus, in the example below, the matrix A is equal to the matrix B but not to the matrix C . A = 3 1 5 4 2 9 - B = 3 1 5 4 2 9 - C = 3 1 9 4 2 5 - Note that the matrix C has the same elements as A, but in a different order . 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. Example 2. First Semester Second Semester Accounting Econometrics Accounting Econometrics Tim 45 65 55 73 Alice 58 60 67 72 Claire 75 70 80 78 We can add the results for both semesters for Accounting for Tim (45 + 55 = 100) and both semester results for Econometrics ( 65 + 73 = 138). A similar process is used for the results for Claire and Alice. If our results are summarized by two matrices, (where the matrix, S 1, represents first semester results and S 2 represents
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