9.4 Matrices 575 Matrix Addition and Groups Consider an infinite set consisting of all matrices that have the same given dimensions— for example, all 2 3 × matrices. This set, along with the operation of matrix addition, is a commutative group. We will illustrate one such commutative group in Example 8. Example 7 A Manufacturing Application The Fancy Frock Company manufactures three types of outfits: a dress, a twopiece 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 threepiece suits. Each dress requires 4 units of material and 1 hour of work 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. Solution Let matrix A represent the number of each type of outfit produced. Now try Exercise 53 Dress Two- piece Three- piece A 20 30 50 = ⎡⎣ ⎤ ⎦ The units of material and time requirements for each outfit are indicated in matrix B. B 4 1 5 2 6 3 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ Dress Two-piece Three-piece Material Hours The product of A and B, or A B, × will give the total number of units of material and the total number of hours of work needed for that day’s production. A B 20 30 50 4 1 5 2 6 3 20(4) 30(5) 50(6) 20(1) 30(2) 50(3) 530 230 × = ⎡⎣ ⎤ ⎦ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = + + + + ⎡ ⎣ ⎤ ⎦ = ⎡⎣ ⎤ ⎦ Thus, a total of 530 units of material and a total of 230 hours of work are needed that day. 7 Profiles in Mathematics James Sylvester, William Rowan Hamilton, and Arthur Cayley Three mathematicians played important roles in the development of matrix theory: James Sylvester (1814–1897), William Rowan Hamilton (1805–1865), and Arthur Cayley (1821–1895). Sylvester and Cayley cooperated in developing matrix theory. Sylvester was the first to use the term matrix . Hamilton, a noted physicist, astronomer, and mathematician, also used what was essentially the algebra of matrices under the name linear and vector functions . The mathematical concept of vector space grew out of Hamilton’s work on the algebra of vectors. Example 8 A Commutative Group Consider the set of all 2 3 × matrices under the operation of matrix addition. Show that this mathematical system is a commutative group. Assume that the associative property holds for matrix addition. Solution For this system to be a commutative group, it must satisfy the five properties of a commutative group discussed on page 546. 1. Closure : The sum of any two 2 3 × matrices is always another 2 3 × matrix. Therefore, the system is closed under the operation of matrix addition.
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