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  • Chapter 4Matrices: Homework

    INTRODUCTION TO MATRICES

    A vendor sells hot dogs and corn dogs at three different locations. His total sales (in hundreds) for January and February from the three locations are given in the table below.

    Table 4.1.
     
    JanuaryFebruary
    Hot DogsCorn DogsHot DogsCorn Dogs
    Place I10887
    Place II8667
    Place III6465

    Represent these tables as 3×2 matrices J and F, and answer Exercise 1., Exercise 2., Exercise 3., and Exercise 4.. problems 1 - 4.

    Determine total sales for the two months, that is, find J+F.

    (4.1)
    _autogen-svg2png-0005.png
    Exercise 2.

    Find the difference in sales, JF.

    If hot dogs sell for $3 and corn dogs for $2, find the revenue from the sale of hot dogs and corn dogs. Hint: Let P be a 2×1 matrix. Find (J+F)P.

    (4.2)
    _autogen-svg2png-0010.png
    Exercise 4.

    If March sales will be up from February by 10%, 15%, and 20% at Place I, Place II, and Place III, respectively, find the expected number of hot dogs, and corn dogs to be sold in March. Hint: Let R be a 1×3 matrix with entries 1.10, 1.15, and 1.20. Find _autogen-svg2png-0013.png.

    Determine the sums and products in the next 5 problems. Given the matrices A, B, C, and D as follows:

    _autogen-svg2png-0018.png _autogen-svg2png-0019.png _autogen-svg2png-0020.png _autogen-svg2png-0021.png

    (4.3)3A−2B
    (4.4)
    _autogen-svg2png-0023.png
    Exercise 6.
    (4.5)
    _autogen-svg2png-0024.png
    (4.6)A2
    (4.7)
    _autogen-svg2png-0026.png
    Exercise 8.
    (4.8)
    _autogen-svg2png-0027.png
    (4.9)
    _autogen-svg2png-0028.png
    (4.10)
    _autogen-svg2png-0029.png
    Exercise 10.
    (4.11)A2B

    Let _autogen-svg2png-0031.png and _autogen-svg2png-0032.png, find _autogen-svg2png-0033.png.

    (4.12)
    _autogen-svg2png-0034.png
    Exercise 12.

    Let _autogen-svg2png-0035.png , find _autogen-svg2png-0036.png.

    Express the following systems as _autogen-svg2png-0037.png, where A, X, and B are matrices.

    (4.13)4x−5y=6
    (4.14)5x−6y=7
    (4.15)
    _autogen-svg2png-0043.png
    Exercise 14.
    (4.16)x−2y+2z=3
    (4.17)x−3y+4z=7
    (4.18)x−2y−3z=−12
    (4.19)2x+3z=17
    (4.20)3x−2y=10
    (4.21)5y+2z=11
    (4.22)
    _autogen-svg2png-0050.png
    Exercise 16.
    (4.23)
    _autogen-svg2png-0051.png

    SYSTEMS OF LINEAR EQUATIONS

    Solve the following by the Gauss-Jordan Method. Show all work.

    (4.24)x+3y=1
    (4.25)2x−5y=13
    (4,-1)
    Exercise 18.
    (4.26)xyz=−1
    (4.27)x−3y+2z=7
    (4.28)2x−y+z=3
    (4.29)x+2y+3z=9
    (4.30)3x+4y+z=5
    (4.31)2x−y+2z=11
    (2, -1, 3)
    Exercise 20.
    (4.32)x+2y=0
    (4.33)y+z=3
    (4.34)x+3z=14

    Two apples and four bananas cost $2.00 and three apples and five bananas cost $2.70. Find the price of each.

    (0.4, 0.3)
    Exercise 22.

    A bowl of corn flakes, a cup of milk, and an egg provide 16 grams of protein. A cup of milk and two eggs provide 21 grams of protein, and two bowls of corn flakes with two cups of milk provide 16 grams of protein. How much protein is provided by one unit of each of these three foods.

    (4.35)x+2y=10
    (4.36)y+z=5
    (4.37)z+w=3
    (4.38)x+w=5
    (4, 3, 2, 1)
    Exercise 24.
    (4.39)x+w=6
    (4.40)2x+y+w=16
    (4.41)x−2z=0
    (4.42)z+w=5

    SYSTEMS OF LINEAR EQUATIONS – SPECIAL CASES

    Solve the following inconsistent or dependent systems by using the Gauss-Jordan method.

    (4.43)2x+6y=8
    (4.44)x+3y=4
    (4.45)(4−3t,t)
    Exercise 26.

    The sum of the digits of a two digit number is 9. The sum of the number and the number obtained by interchanging the digits is 99. Find the number.

    (4.46)2x−y=10
    (4.47)−4x+2y=15
    Inconsistent system, no solution
    Exercise 28.
    (4.48)x+y+z=6
    (4.49)3x+2y+z=14
    (4.50)4x+3y+2z=20
    (4.51)x+2y−4z=1
    (4.52)2x−3y+8z=9
    (3−4/7t,−1+16/7t,t)
    Exercise 30.

    Jessica has a collection of 15 coins consisting of nickels, dimes and quarters. If the total worth of the coins is $1.80, how many are there of each? Find all three solutions.

    The latest reports indicate that there are altogether 20,000 American, French, and Russian troops in Bosnia. The sum of the number of Russian troops and twice the American troops equals 10,000. Furthermore, the Americans have 5,000 more troops than the French. Are these reports consistent?

    No, they are not consistent.
    Exercise 32.
    (4.53)x+y+2z=0
    (4.54)x+2y+z=0
    (4.55)2x+3y+3z=0

    Find three solutions to the following system of equations.

    (4.56)x+2y+z=12
    (4.57)y=3
    (5, 3, 1), (4, 3, 2) (3, 3, 3)
    Exercise 34.

    For what values of k the following system of equations have a). No solution? b). Infinitely many solutions?

    (4.58)x+2y=5
    (4.59)2x+4y=k
    (4.60)x+3y−z=5
    ( 5−3s+t, s, t)
    Exercise 36.

    Why is it not possible for a linear system to have exactly two solutions? Explain geometrically.

    INVERSE MATRICES

    In the next two problems, verify that the given matrices are inverses of each other.

    Exercise 37.
    (4.61)
    _autogen-svg2png-0094.png
    Exercise 38.
    (4.62)
    _autogen-svg2png-0095.png

    In the following problems, find the inverse of each matrix by the row-reduction method.

    (4.63)
    _autogen-svg2png-0096.png
    (4.64)
    _autogen-svg2png-0097.png
    Exercise 40.
    (4.65)
    _autogen-svg2png-0098.png
    (4.66)
    _autogen-svg2png-0099.png
    (4.67)
    _autogen-svg2png-0100.png
    Exercise 42.
    (4.68)
    _autogen-svg2png-0101.png

    In the following problems, first express the system as AX = B, and then solve using matrix inverses found in the preceding four problems.

    (4.69)3x−5y=2
    (4.70)x+2y=0
    (4, 2)
    Exercise 44.
    (4.71)
    _autogen-svg2png-0104.png
    (4.72)
    _autogen-svg2png-0105.png
    (3, 3, 4)
    Exercise 46.
    (4.73)
    _autogen-svg2png-0106.png

    Why is it necessary that a matrix be a square matrix for its inverse to exist? Explain by relating the matrix to a system of equations.

    If a matrix M has an inverse, then the system of linear equations that has M as its coefficient matrix has a unique solution. If a system of linear equations has a unique solution, then the number of equations must be the same as the number of variables. Therefore, the matrix that represents its coefficient matrix must be a square matrix.
    Exercise 48.

    Suppose we are solving a system _autogen-svg2png-0109.png by the matrix inverse method, but discover A has no inverse. How else can we solve this system? What can be said about the solutions of this system?

    APPLICATION OF MATRICES IN CRYPTOGRAPHY

    In the following problems, the letters A to Z correspond to the numbers 1 to 26, as shown below, and a space is represented by the number 27.

    Table 4.2.
    ABCDEFGHIJKLM
    12345678910111213
    NOPQRSTUVWXYZ
    14151617181920212223242526

    In the next two problems, use the matrix A, given below, to encode the given messages.

    _autogen-svg2png-0111.png

    In the two problems following, decode the messages that were encoded using matrix A.

    Make sure to consider the spaces between words, but ignore all punctuation. Add a final space if necessary.

    Encode the message: WATCH OUT!

    (4.74)
    _autogen-svg2png-0112.png
    Exercise 50.

    Encode the message: HELP IS ON THE WAY.

    Decode the following message:

    64 23 102 41 82 32 97 35 71 28 69 32

    RETURN HOME
    Exercise 52.

    Decode the following message:

    105 40 117 48 39 19 69 32 72 27 37 15 114 47

    In the next two problems, use the matrix B, given below, to encode the given messages.

    _autogen-svg2png-0113.png

    In the two problems following, decode the messages that were encoded using matrix B.

    Make sure to consider the spaces between words, but ignore all punctuation. Add a final space(s) if necessary.

    Encode the message using matrix B:

    LUCK IS ON YOUR SIDE.

    (4.75)
    _autogen-svg2png-0116.png
    Exercise 54.

    Encode the message using matrix B:

    MAY THE FORCE BE WITH YOU.

    Decode the following message that was encoded using matrix B:

    8 23 7 4 47 –2 15 102 –12 20 58 15 27 80 18 12 74 –7

    HEAD FOR THE HILLS
    Exercise 56.

    Decode the following message that was encoded using matrix B:

    12 69 –3 11 53 9 5 46 –10 18 95 –9 25 107 4 27 76 22 1 72 –26

    APPLICATIONS – LEONTIEF MODELS

    Solve the following homogeneous system.

    (4.76)x+y+z=0
    (4.77)3x+2y+z=0
    (4.78)4x+3y+2z=0
    ( t, −2t, t)
    Exercise 58.

    Solve the following homogeneous system.

    (4.79)xyz=0
    (4.80)x−3y+2z=0
    (4.81)2x−4y+z=0

    Chris and Ed decide to help each other by doing repairs on each others houses. Chris is a carpenter, and Ed is an electrician. Chris does carpentry work on his house as well as on Ed's house. Similarly, Ed does electrical repairs on his house and on Chris' house. When they are all finished they realize that Chris spent 60% of his time on his own house, and 40% of his time on Ed's house. On the other hand Ed spent half of his time on his house and half on Chris's house. If they originally agreed that each should get about a $1000 for their work, how much money should each get for their work?

    Chris=$1250, Ed=$1,000
    Exercise 60.

    Chris, Ed, and Paul decide to help each other by doing repairs on each others houses. Chris is a carpenter, Ed is an electrician, and Paul is a plumber. Each does work on his own house as well as on the others houses. When they are all finished they realize that Chris spent 30% of his time on his own house, 40% of his time on Ed's house, and 30% on Paul's house. Ed spent half of his time on his own house, 30% on Chris' house, and remaining on Paul's house. Paul spent 40% of the time on his own house, 40% on Chris' house, and 20% on Ed's house. If they originally agreed that each should get about a $1000 for their work, how much money should each get for their work?

    Given the internal consumption matrix A, and the external demand matrix D as follows.

    (4.82)
    _autogen-svg2png-0131.png

    Solve the system using the open model: _autogen-svg2png-0132.png or X=(I−A)−1D

    (4.83)
    _autogen-svg2png-0134.png
    Exercise 62.

    Given the internal consumption matrix A, and the external demand matrix D as follows.

    (4.84)
    _autogen-svg2png-0137.png

    Solve the system using the open model: _autogen-svg2png-0138.png or X=(IA)−1D

    An economy has two industries, farming and building. For every $1 of food produced, the farmer uses $.20 and the builder uses $.15. For every $1 worth of building, the builder uses $.25 and the farmer uses $.20. If the external demand for food is $100,000, and for building $200,000, what should be the total production for each industry in dollars?

    (4.85)Farming=$201,754.38,Building=$307,017.54
    Exercise 64.

    An economy has three industries, farming, building, and clothing. For every $1 of food produced, the farmer uses $.20, the builder uses $.15, and the tailor $.05. For every $1 worth of building, the builder uses $.25, the farmer uses $.20, and the tailor $.10. For every $1 worth of clothing, the tailor uses $.10, the builder uses $.20, the farmer uses $.15. If the external demand for food is $100 million, for building $200 million, and for clothing $300 million, what should be the total production for each in dollars?

    Suppose an economy consists of three industries F, C, and T. The following table gives information about the internal use of each industry's production and external demand in dollars.

    Table 4.3.
      F C T DemandTotal
    F 30102040100
    C 20302050120
    T 10103060110

    Find the proportion of the amounts consumed by each of the industries; that is, find the matrix A.

    (4.86)
    _autogen-svg2png-0151.png
    Exercise 66.

    If in the preceding problem, the consumer demand for F, C, and T becomes 60, 80, and 100, respectively, find the total output and the internal use by each industry to meet that demand.

    CHAPTER REVIEW

    To reinforce her diet, Mrs. Tam bought a bottle containing 30 tablets of Supplement A and a bottle containing 50 tablets of Supplement B. Each tablet of supplement A contains 1000 mg of calcium, 400 mg of magnesium, and 15 mg of zinc, and each tablet of supplement B contains 800 mg of calcium, 500 mg of magnesium, and 20 mg of zinc.

    1. Represent the amount of calcium, magnesium and zinc in each tablet as a 2×3 matrix.

    2. Represent the number of tablets in each bottle as a row matrix.

    3. Use matrix multiplication to determine the total amount of calcium, magnesium, and zinc in both bottles.

    1. _autogen-svg2png-0160.png

    2. _autogen-svg2png-0161.png

    Let matrix _autogen-svg2png-0162.png and _autogen-svg2png-0163.png . Find the following.

    1. _autogen-svg2png-0164.png

    2. 3A−2B

    1. _autogen-svg2png-0166.png

    2. (4.87)
      _autogen-svg2png-0167.png

    Let matrix _autogen-svg2png-0168.png and _autogen-svg2png-0169.png. Find the following.

    1. 2(CD)

    2. C−3D

    1. _autogen-svg2png-0172.png

    2. (4.88)
      _autogen-svg2png-0173.png

    Let matrix _autogen-svg2png-0174.png and _autogen-svg2png-0175.png. Find the following.

    1. _autogen-svg2png-0176.png

    2. _autogen-svg2png-0177.png

    1. _autogen-svg2png-0178.png

    2. (4.89)
      _autogen-svg2png-0179.png

    Let matrix _autogen-svg2png-0180.png and _autogen-svg2png-0181.png . Find the following.

    1. _autogen-svg2png-0182.png

    2. _autogen-svg2png-0183.png

    1. _autogen-svg2png-0184.png

    2. (4.90)
      _autogen-svg2png-0185.png

    Solve the following systems using the Gauss-Jordan Method.

    1. _autogen-svg2png-0186.png

    2. _autogen-svg2png-0187.png

    1. (2, 1, –1)

    2. (3, 2, 1)

    An apple, a banana and three oranges or two apples, two bananas, and an orange, or four bananas and two oranges cost $2. Find the price of each.

    Apple = $.50; banana = $.30; orange = $.40

    Solve the following systems. If a system has an infinite number of solutions, first express the solution in parametric form, and then determine one particular solution.

    1. _autogen-svg2png-0188.png

    2. _autogen-svg2png-0189.png

    1. x=6−t, y=0, z=t; (5, 0, 1)

    2. no solution

    Elise has a collection of 12 coins consisting of nickels, dimes and quarters. If the total worth of the coins is $1.80, how many are there of each? Find all possible solutions.

    n=3t−12, d=−4t+24, q=t; n=3, d=4, q=5

    Solve the following systems. If a system has an infinite number of solutions, first express the solution in parametric form, and then find a particular solution.

    1. _autogen-svg2png-0199.png

    2. _autogen-svg2png-0200.png

    1. x=4−2t, y=t, z=3; (4, 0, 3)

    2. x=5−4t, y=2−t, z=t; (1, 1, 1)

    Solve the following systems. If a system has an infinite number of solutions, first express the solution in parametric form, and then provide one particular solution.

    1. _autogen-svg2png-0207.png

    2. _autogen-svg2png-0208.png

    1. x=.5t, y=t, z−2t; (1, 2, 2)

    2. no solution

    Find the inverse of the following matrices.

    1. _autogen-svg2png-0212.png

    2. _autogen-svg2png-0213.png

    1. (4.91)
      _autogen-svg2png-0214.png

    2. (4.92)
      _autogen-svg2png-0215.png

    Solve the following systems using the matrix inverse method.

    1. _autogen-svg2png-0216.png

    2. _autogen-svg2png-0217.png

    1. (-1, 4, 2)

    2. (6, 4, 2, -1)

    Use matrix A, given below, to encode the following messages. The space between the letters is represented by the number 27, and all punctuation is ignored.

    (4.93)
    _autogen-svg2png-0219.png
    1. TAKE IT AND RUN.

    2. GET OUT QUICK.

    1. (4.94)
      _autogen-svg2png-0220.png

    2. (4.95)
      _autogen-svg2png-0221.png

    Decode the following messages that were encoded using matrix A in the above problem.

    1. 44, 71, 15, 18, 27, 1, 68, 82, 27, 69, 76, 27, 19, 33, 9

    2. 37, 64, 15, 36, 54, 15, 67, 75, 20, 59, 66, 27, 39, 43, 12

    1. NO PAIN NO GAIN

    2. GO FOR THE GOLD

    Chris, Bob, and Matt decide to help each other study during the final exams. Chris's favorite subject is chemistry, Bob loves biology, and Matt knows his math. Each studies his own subject as well as helps the others learn their subjects. After the finals, they realize that Chris spent 40% of his time studying his own subject chemistry, 30% of his time helping Bob learn chemistry, and 30% of the time helping Matt learn chemistry. Bob spent 30% of his time studying his own subject biology, 30% of his time helping Chris learn biology, and 40% of the time helping Matt learn biology. Matt spent 20% of his time studying his own subject math, 40% of his time helping Chris learn math, and 40% of the time helping Bob learn math. If they originally agreed that each should work about 33 hours, how long did each work?

    (4.96)x=40/33t,y=36/33t,z=t;Chris=40hrs,Bob=36hrs,Matt=33hrs

    As in the previous problem, Chris, Bob, and Matt decide to not only help each other study during the final exams, but also tutor others to make a little money. Chris spends 30% of his time studying chemistry, 15% of his time helping Bob with chemistry, and 25% helping Matt with chemistry. Bob spends 25% of his time studying biology, 15% helping Chris with biology, and 30% helping Matt. Similarly, Matt spends 20% of his time on his own math, 20% helping Chris, and 20% helping Bob. If they spend respectively, 12, 12, and 10 hours tutoring others, how many total hours are they going to end up working?

    (4.97)Chris=34.1hrs,Bob=32.2hrs,Matt=35.2hrs
    Solutions