570 CHAPTER 9 Mathematical Systems Columns Freshmen Sophomores Juniors Seniors Rows In favor 102 93 22 35 Opposed 82 94 23 49 The numbers, symbols, or expressions in the rows and columns of a matrix are called the elements of the matrix. The matrix given above contains eight elements. The dimensions of a matrix may be indicated with the notation m n, × where m is the number of rows and n is the number of columns in the matrix. Because the given matrix has 2 rows and 4 columns, it is a 2 by 4, written 2 4, × matrix. In this text, from this point onward, we use brackets, [ ], to indicate a matrix. Consider the two matrices below. A matrix that contains the same number of rows and columns is called a square matrix . Following is an example of a 2 2 × square matrix and a 3 3 × square matrix. ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 2 3 5 2 4 6 1 2 3 0 5 2 1 Two matrices are equal if and only if they have the same elements in the same relative positions. We use capital letters to represent matrices. Suppose we are given the following matrices A and B and that A B. = What are the values of x and y? = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ A B x y 2 5 7 1 , 5 7 Since the matrices are equal, the corresponding elements must be the same, so x 2 = and y 1. = − MATHEMATICS TODAY Matrices Are Everywhere You are already familiar with matrices, although you may not be aware of it. A matrix is a good way to display numerical data, as illustrated on this trail sign in Yosemite National Park. Why This Is Important Most charts and tables you see daily, including spreadsheets, are matrices. Addition of Matrices Two matrices can be added only if they have the same dimensions; that is, the same number of rows and the same number of columns. To obtain the sum of two matrices with the same dimensions, add the corresponding elements of the two matrices. Matrix Addition Example 1 Adding Matrices Determine A B + if A 5 2 3 8 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ and B 0 2 4 7 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Solution A B 5 2 3 8 0 2 4 7 5 0 2 2 3 4 8 7 5 4 1 15 + = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + + − + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 7 Now try Exercise 9 Example 2 Sales of Bicycles Towpath Bicycle Corporation owns and operates two stores, one in Virginia and one in North Carolina. The number of mountain bicycles (MB) and racing bicycles (RB) sold in each store during January through June and during July through December are indicated in the matrices that follow. We will call the matrices A and B.
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