Transcript matrix
Section 7.3 Matrices Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Introduction to Matrices Addition of Matrices Subtraction of Matrices Multiplying a Matrix by a Real Number Multiplication of Matrices 7.3-2 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Matrix A matrix is a rectangular array of elements. An array is a systematic arrangement of numbers or symbols in rows and columns. 7.3-3 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Matrix Matrices (the plural of matrix) may be used to display information and to solve systems of linear equations. The numbers in the rows and columns of a matrix are called the elements of the matrix. 7.3-4 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Dimensions of a Matrix The dimensions of a matrix may be indicated with the notation r × s, where r is the number of rows and s is the number of columns of a matrix. This is a 2 × 4 matrix. 102 93 22 35 82 94 23 49 7.3-5 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Dimensions of a Matrix A matrix that contains the same number of rows and columns is called a square matrix. 2×2 square matrix 2 3 5 2 7.3-6 3×3 square matrix 4 0 2 4 1 3 6 8 9 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Equal Matrices Two matrices are equal if and only if they have the same elements in the same relative positions. 4 0 2 4 0 2 4 1 3 4 1 3 6 8 9 6 8 9 7.3-7 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Addition and Subtraction of Matrices Two matrices can only be added or subtracted if they have the same dimensions. The corresponding elements of the two matrices are either added or subtracted. 7.3-8 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: Subtracting Matrices Determine A – B if 3 6 2 4 A and B 8 3 5 1 Solution 3 6 2 4 AB 5 1 8 3 3 2 6 (4) 1 10 5 8 1 (3) 3 2 7.3-9 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Multiplying a Matrix by a Real Number A matrix may be multiplied by a real number, a scalar, by multiplying each entry in the matrix by the real number. e f A h i 7.3-10 g j 3e 3f 3A 3h 3i Copyright 2013, 2010, 2007, Pearson, Education, Inc. 3g 3 j Multiplication of Matrices Multiplication of matrices is possible only when the number of columns in the first matrix is the same as the number of rows of the second matrix. For example, we can multiply a 2 × 2 matrix times a 2 × 3 matrix. 7.3-11 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Multiplication of Matrices In general, a b e f AB c d h i g j ae bh af bi ag bj ce dh cf di cg dj Note that Matrix Multiplication is not commutative: A × B ≠ B × A. 7.3-12 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 7: Multiplying Matrices Determine A × B if 5 1 3 3 1 A and B 4 2 2 8 0 Solution 3 1 5 1 3 AB 4 2 2 8 0 3(5) 1(2) 3(1) 1(8) 3(3) 1(0) 4(5) 2(2) 4(1) 2(8) 4(3) 2(0) 7.3-13 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Multiplicative Identity Matrix Square matrices have a multiplicative identity matrix. Note that in any multiplicative identity matrix, 1’s go diagonally from top left to bottom right and all other elements in the matrix are 0’s. 1 0 0 1 0 I 0 1 0 I 0 0 1 0 1 7.3-14 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Multiplicative Identity Matrix For any square matrix A, A × I = I × A = A. 4 3 If A then, 2 1 4 3 1 0 4 3 A I A 2 1 0 1 2 1 7.3-15 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: A Manufacturing Application The Fancy Frock Company manufactures three types of women’s outfits: a dress, a two-piece suit (skirt and jacket), and a three-piece suit (skirt, jacket, and a vest). On a particular day, the firm produces 20 dresses, 30 two-piece suits, and 50 three-piece suits. Each dress requires 4 units of material and 1 hour of work 7.3-16 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: A Manufacturing Application to produce, each two-piece suit requires 5 units of material and 2 hours of work to produce, and each three-piece suit requires 6 units of material and 3 hours to produce. Use matrix multiplication to determine the total number of units of material and the total number of hours needed for that day’s production. 7.3-17 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: A Manufacturing Application Solution Two- ThreeDress piece piece A 20 30 50 Material Hours 4 1 B 5 2 6 3 7.3-18 Dress Two-piece Three-piece Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: A Manufacturing Application Solution The product of A and B, will give the total number of units of material and the total number of hours of work needed for that day’s production. 4 1 A B 20 30 50 5 2 6 3 7.3-19 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9: A Manufacturing Application 4 1 Solution A B 20 30 50 5 2 6 3 20(4) 30(5) 50(6) 20(1) 30(2) 50(3) 530 230 A total of 530 units of material and a total of 230 hours of work are needed that day. 7.3-20 Copyright 2013, 2010, 2007, Pearson, Education, Inc.