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1 There are 300 adult people in a room, none of them obese Explain why two of them must have the same weight (in whole numbers of pounds)
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2 There are 50 people in a room, none of them obese Explain why two of them must have the same waist measurement (in whole numbers of inches) 3 Explain why the answer to Exercise 2 changes if the waist measurement is changed to whole numbers of millimeters 4 There are 20 people sitting in a waiting room The functionary in charge must choose ve of these people to go to the green sanctuary and three of these people to go to the red sanctuary In how many different ways can she do this 5 In a standard deck of 52 playing cards, in how many different ways can you form two-of-a-kind 6 In a standard deck of 52 playing cards, in how many different ways can you form four-of-a-kind 7 A standard die used for gambling is a six-sided cube, with the sides numbered 1 through 6 You usually roll two dice at a time, and the two face-up values are added together to give your score What is the likelihood that you will roll a seven 8 Refer to the last exercise for terminology What is the chance that you will roll two dice and get a two How about a 12 Are there any other values that give this same answer Why or why not 9 Again refer to Exercise 7 for terminology Now suppose that you are rolling three dice Your score is obtained by adding together the three face values What is your probability of getting a 10 10 Solve the recursion a0 = 3, a1 = 5, and a j = a j 1 + 2a j 2 for j 2 11 Suppose we take a nite collection of points in the plane and connect every pair with an edge We color each edge either red or blue or green Will there be a triangle of just one color With 16 points the answer is no but with 17 points the answer is yes Discuss
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71 What Is a Matrix
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A matrix is a rectangular array of numbers or variables or other algebraic objects An example of a matrix is a f k p b g l q c h m r d i n s e j o t
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We call this a 4 5 matrix because it has four rows and ve columns In general, an m n matrix has m rows and n columns When Arthur Cayley (1821 1895) invented matrices in the late nineteenth century, he bragged that he invented something that was of no earthly use Rarely has a person s assessment of his own work been more inaccurate Today matrices are used in all parts of mathematics, in engineering and physics, in the social sciences, in statistics, and in any part of analytical thought where it is necessary to keep track
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of (and to manipulate) information What is important about matrices is that they can be combined in a number of useful ways addition, multiplication, inversion, composition, and others and each of these operations has signi cance for the information that the matrix contains We shall learn a bit about these ideas in the present chapter
72 Fundamental Operations on Matrices
We typically denote a matrix by a capital roman letter like A or M The elements of the matrix A are designated by aij , where i is the row in which the element is located and j is the column in which the element is located To take a speci c example, consider the matrix 3 1 4 2 6 5 4 0 A= 1 9 14 8 For this matrix, a23 = 4 because the element of the matrix that is in the second row and third column is 4 Likewise, a32 = 9 and a33 = 14 Notice that, for this particular matrix, there are elements aij for 1 i 3 (because there are three rows) and for 1 j 4 (because there are four columns) We can add two matrices only when they have the same size If A = (aij ) is an m n matrix and B = (bij ) is another m n matrix, then their sum is A + B = (aij + bij ) or A + B = (aij ) + (bij ) = (aij + bij ) EXAMPLE 71 As a concrete illustration of matrix addition, let 3 6 2 4 A= 1 0 Then 3 2 A+B = 1 5 6 4 + 1 0 0 2 3 1 6 = 1 11 3 2 11 5 B= 1 0 3 6 11
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