Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. 2.4 Matrices A matrix is an ordered rectangular array of numbers. A matrix with m rows and n columns has size m x n. The entry in the ith row and jth column is denoted by aij. Ex. 3 Rows 4 Columns Size = Row x Column = 3 x 4 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Square matrix – same number of rows as columns. Ex. Here is a 2 x 2 matrix: Two matrices are equal if they have the same size and their corresponding entries are equal. Ex. Find x and y. Corresponding entries are equal y + 1 = 4 and = 7 y = 3 and x = 14 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Addition and Subtraction of Matrices If A and B are two matrices of the same size, then The sum A + B is found by adding corresponding entries in the two matrices. The difference A – B is found by subtracting the corresponding entries in B and A. Also, we have the Commutative law: A + B = B + A and Associative law (A + B) + C = A + (B + C) for addition. A zero matrix is one in which all entries are zero. The zero Matrix O has the property that A+O=O+A=A Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Given matrices A and B, find A + B and A – B. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Transpose of a Matrix Transpose of a Matrix – If A is an m x n matrix with elements aij, then the transpose of A is the n x m matrix AT with elements aji. Example. 1 4 7 2 5 8 3 6 9 Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Scalar Product – If A is a matrix and c is a real number, then the scalar product cA is the matrix obtained by multiplying each entry of A by c. Example. Given the matrix find 5A. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. The market share of motorcycles in the United States in 2001 follows: Honda 27.9%, Harley-Davidson 21.9%, Yamaha 19.2%, Yamaha 19.2%, Suzuki 11%, Kawasaki 9.1%, and others 10.9%. The corresponding figures for 2002 are 27.6%, 23.3%, 18.2%, 10.5%, 8.8%, and 11.6%, respectively. Express this information using a 2x6 matrix. What is the sum of all the elements in the first row? In the second row? Is this expected? Which company gained the most market share between 2001 and 2002? Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Solution: The sum of all elements in the first row is 100%. The sum of all elements in the second row is 100%. Harley-Davidson gained the most: 23.3-21.9 or 1.4%. Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.