Copyright © Cengage Learning. All rights reserved. 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES Warm Up Read pp up to and including the box labeled “Matrix”.

Copyright © Cengage Learning. All rights reserved. 2.4 Matrices

3 Using Matrices to Represent Data

4 The Acrosonic Company manufactures four different loudspeaker systems at three separate locations. The company’s May output is described in Table 1. Table 1

5 Using Matrices to Represent Data Now, if we agree to preserve the relative location of each entry in Table 1, we can summarize the set of data as follows: The array of numbers displayed here is an example of a matrix. A matrix summarizing the data in Table 1

6 Using Matrices to Represent Data Observe that the numbers in row 1 give the output of models A, B, C, and D of Acrosonic loudspeaker systems manufactured at Location I; similarly, the numbers in rows 2 and 3 give the respective outputs of these loudspeaker systems at Locations II and III. The numbers in each column of the matrix give the outputs of a particular model of loudspeaker system manufactured at each of the company’s three manufacturing locations.

7 Using Matrices to Represent Data The real numbers that make up the array are called the entries, or elements, of the matrix. The entries in a row in the array are referred to as a row of the matrix, whereas the entries in a column in the array are referred to as a column of the matrix. Matrix A, for example, has two rows and three columns, which may be identified as follows: A 2  3 matrix

8 Using Matrices to Represent Data The size, or dimension, of a matrix is described in terms of the number of rows and columns of the matrix. For example, matrix A has two rows and three columns and is said to have size 2 by 3, denoted 2  3. In general, a matrix having m rows and n columns is said to have size m  n.

9 Using Matrices to Represent Data A matrix of size 1  n—a matrix having one row and n columns—is referred to as a row matrix, or row vector, of dimension n. For example, the matrix D is a row vector of dimension 4. Similarly, a matrix having m rows and one column is referred to as a column matrix, or column vector, of dimension m. The matrix C is a column vector of dimension 4. Finally, an n  n matrix—that is, a matrix having the same number of rows as columns—is called a square matrix.

10 Using Matrices to Represent Data For example, the matrix is a square matrix of size 3  3, or simply of size 3. A 3  3 square matrix

11 Equality of Matrices

12 Equality of Matrices Two matrices are said to be equal if they have the same size and their corresponding entries are equal.

13 Example 2 Solve the following matrix equation for x, y, and z: Solution: Since the corresponding elements of the two matrices must be equal, we find that x = 4, z = 3, and y – 1 = 1, or y = 2.

14 Practice p. 109 Self-Check Exercises #2