800 CHAPTER 11 Systems of Equations and Inequalities DEFINITION Matrix Multiplication Let A denote an m by r matrix and B denote an r by n matrix. The product AB is defined as the m by n matrix whose entry in row i, column j is the product of the i th row of A and the j th column of B. In Words To find the product AB, the number of columns in the left matrix A must equal the number of rows in the right matrix B. Using Matrices to Compute Revenue A clothing store sells men’s shirts for $40, silk ties for $20, and wool suits for $400. Last month, the store sold 100 shirts, 200 ties, and 50 suits.What was the total revenue from these sales? EXAMPLE 7 Solution Set up a row vector R to represent the prices of these three items and a column vector C to represent the corresponding number of items sold. Then [ ] = = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ R C 40 20 400 100 200 50 The total revenue from these sales equals the product RC. That is, [ ] = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ = ⋅ + ⋅ + ⋅ = RC 40 20 400 100 200 50 40 100 20 200 400 50 $28,000 Shirts Prices Ties Suits Number sold Shirts Ties Suits Shirt revenue Tie revenue Suit revenue Total revenue The definition for multiplying two matrices is based on the definition of a row vector times a column vector. The definition of the product AB of two matrices A and B, in this order, requires that the number of columns of A equals the number of rows of B; otherwise, the product is not defined. A m r B r n by by Must be same for AB to be defined AB is m by n. An example will help clarify the definition. Multiplying Two Matrices Find the product AB if = ⎡ − ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ = − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ A B 2 4 1 5 8 0 and 2 5 1 4 4 8 0 6 3 1 2 1 EXAMPLE 8 Solution First, observe that A is 2 by 3 and B is 3 by 4. The number of columns in A equals the number of rows in B, so the product AB is defined and will be a 2 by 4 matrix.
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