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Matrices: Homework

Module by: UniqU, LLC. E-mail the author

Based on: Applied Finite Mathematics: Chapter 02 by Rupinder Sekhon

Summary: This chapter covers principles of matrices. After completing this chapter students should be able to: complete matrix operations; solve linear systems using Gauss-Jordan method; Solve linear systems using the matrix inverse method and complete application problems.

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 1
 
January February
Hot Dogs Corn Dogs Hot Dogs Corn Dogs
Place I 10 8 8 7
Place II 8 6 6 7
Place III 6 4 6 5

Represent these tables as 3×23×2 size 12{3 times 2} {} matrices JJ size 12{J} {} and FF size 12{F} {}, and answer Exercise 1, Exercise 2, Exercise 3, and Exercise 4. problems 1 - 4.

Exercise 1

Determine total sales for the two months, that is, find J+FJ+F size 12{J+F} {}.

Solution

1815141312918151413129 size 12{ left [ matrix { "18" {} # "15" {} ## "14" {} # "13" {} ## "12" {} # 9{} } right ]} {}
(1)

Exercise 2

Find the difference in sales, J−FJ−F size 12{J - F} {}.

Exercise 3

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 PP size 12{P} {} be a 2×12×1 size 12{2 times 1} {} matrix. Find J+FPJ+FP size 12{ left (J+F right )P} {}.

Solution

846854846854 size 12{ left [ matrix { "84" {} ## "68" {} ## "54" } right ]} {}
(2)

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 RR size 12{R} {} be a 1×31×3 size 12{1 times 3} {} matrix with entries 1.10, 1.15, and 1.20. Find RFRF size 12{ ital "RF"} {}.

Determine the sums and products in the next 5 problems. Given the matrices AA size 12{A} {}, BB size 12{B} {}, CC size 12{C} {}, and DD size 12{D} {} as follows:

A = 3 6 1 0 1 3 2 4 1 A = 3 6 1 0 1 3 2 4 1 size 12{A= left [ matrix { 3 {} # 6 {} # 1 {} ## 0 {} # 1 {} # 3 {} ## 2 {} # 4 {} # 1{} } right ]} {} B = 1 − 1 2 1 4 2 3 1 1 B = 1 − 1 2 1 4 2 3 1 1 size 12{B= left [ matrix { 1 {} # - 1 {} # 2 {} ## 1 {} # 4 {} # 2 {} ## 3 {} # 1 {} # 1{} } right ]} {} C = 1 2 3 C = 1 2 3 size 12{C= left [ matrix { 1 {} ## 2 {} ## 3 } right ]} {} D = 2 3 2 D = 2 3 2 size 12{D= left [ matrix { 2 {} # 3 {} # 2{} } right ]} {}

Exercise 5

3A−2B3A−2B size 12{3A - 2B} {}
(3)

Solution

720−1−2−550101720−1−2−550101 size 12{ left [ matrix { 7 {} # "20" {} # - 1 {} ## - 2 {} # - 5 {} # 5 {} ## 0 {} # "10" {} # 1{} } right ]} {}
(4)

Exercise 6

AB+BAAB+BA size 12{ ital "AB"+ ital "BA"} {}
(5)

Exercise 7

A2A2 size 12{A2} {}
(6)

Solution

112822613682015112822613682015 size 12{ left [ matrix { "11" {} # "28" {} # "22" {} ## 6 {} # "13" {} # 6 {} ## 8 {} # "20" {} # "15"{} } right ]} {}
(7)

Exercise 8

2BC2BC size 12{2 ital "BC"} {}
(8)

Exercise 9

2CD+3AB2CD+3AB size 12{2 ital "CD"+3 ital "AB"} {}
(9)

Solution

407261383323396351407261383323396351 size 12{ left [ matrix { "40" {} # "72" {} # "61" {} ## "38" {} # "33" {} # "23" {} ## "39" {} # "63" {} # "51"{} } right ]} {}
(10)

Exercise 10

A2BA2B size 12{A2B} {}
(11)

Exercise 11

Let E=mnpqE=mnpq size 12{E= left [ matrix { m {} # n {} ## p {} # q{} } right ]} {} and F=abcdF=abcd size 12{F= left [ matrix { a {} # b {} ## c {} # d{} } right ]} {}, find EFEF size 12{ ital "EF"} {}.

Solution

ma+ncmb+ndpa+qcpb+qdma+ncmb+ndpa+qcpb+qd size 12{ left [ matrix { ital "ma"+ ital "nc" {} # ital "mb"+ ital "nd" {} ## ital "pa"+ ital "qc" {} # ital "pb"+ ital "qd"{} } right ]} {}
(12)

Exercise 12

Let G=361013241H=xyzG=361013241 size 12{G= left [ matrix { 3 {} # 6 {} # 1 {} ## 0 {} # 1 {} # 3 {} ## 2 {} # 4 {} # 1{} } right ]} {}H=xyz size 12{H= left [ matrix { x {} ## y {} ## z } right ]} {} , find GHGH size 12{ ital "GH"} {}.

Express the following systems as AX=BAX=B size 12{ ital "AX"=B} {}, where AA size 12{A} {}, XX size 12{X} {}, and BB size 12{B} {} are matrices.

Exercise 13

4x−5y=64x−5y=6 size 12{4x - 5y=6} {}
(13)
5x−6y=75x−6y=7 size 12{5x - 6y=7} {}
(14)

Solution

4−55−6xy=674−55−6xy=67 size 12{ left [ matrix { 4 {} # - 5 {} ## 5 {} # - 6{} } right ] left [ matrix { x {} ## y } right ]= left [ matrix { 6 {} ## 7 } right ]} {}
(15)

Exercise 14

x−2y+2z=3x−2y+2z=3 size 12{x - 2y+2z=3} {}
(16)
x−3y+4z=7x−3y+4z=7 size 12{x - 3y+4z=7} {}
(17)
x−2y−3z=−12x−2y−3z=−12 size 12{x - 2y - 3z= - "12"} {}
(18)

Exercise 15

2x+3z=172x+3z=17 size 12{2x+3z="17"} {}
(19)
3x−2y=103x−2y=10 size 12{3x - 2y="10"} {}
(20)
5y+2z=115y+2z=11 size 12{5y+2z="11"} {}
(21)

Solution

2033−20052xyz=1710112033−20052xyz=171011 size 12{ left [ matrix { 2 {} # 0 {} # 3 {} ## 3 {} # - 2 {} # 0 {} ## 0 {} # 5 {} # 2{} } right ] left [ matrix { x {} ## y {} ## z } right ]= left [ matrix { "17" {} ## "10" {} ## "11" } right ]} {}
(22)

Exercise 16

x+2y+3z+2w=14x−2y−z=−5y−2z+4w=9x+3z+3w=15x+2y+3z+2w=14x−2y−z=−5y−2z+4w=9x+3z+3w=15 size 12{ matrix { x {} # +{} {} # 2y {} # +{} {} # 3z {} # +{} {} # 2w {} # ={} {} # "14" {} ## x {} # - {} {} # 2y {} # - {} {} # z {} # {} # {} # ={} {} # - 5 {} ## y {} # - {} {} # 2z {} # {} # {} # +{} {} # 4w {} # ={} {} # 9 {} ## x {} # +{} {} # 3z {} # {} # {} # +{} {} # 3w {} # ={} {} # "15"{} } } {}
(23)

SYSTEMS OF LINEAR EQUATIONS

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

Exercise 17

x+3y=1x+3y=1 size 12{x+3y=1} {}
(24)
2x−5y=132x−5y=13 size 12{2x - 5y="13"} {}
(25)

Solution

(4,-1)

Exercise 18

x−y−z=−1x−y−z=−1 size 12{x - y - z= - 1} {}
(26)
x−3y+2z=7x−3y+2z=7 size 12{x - 3y+2z=7} {}
(27)
2x−y+z=32x−y+z=3 size 12{2x - y+z=3} {}
(28)

Exercise 19

x+2y+3z=9x+2y+3z=9 size 12{x+2y+3z=9} {}
(29)
3x+4y+z=53x+4y+z=5 size 12{3x+4y+z=5} {}
(30)
2x−y+2z=112x−y+2z=11 size 12{2x - y+2z="11"} {}
(31)

Solution

(2, -1, 3)

Exercise 20

x+2y=0x+2y=0 size 12{x+2y=0} {}
(32)
y+z=3y+z=3 size 12{y+z=3} {}
(33)
x+3z=14x+3z=14 size 12{x+3z="14"} {}
(34)

Exercise 21

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

Solution

(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.

Exercise 23

x+2y=10x+2y=10 size 12{x+2y="10"} {}
(35)
y+z=5y+z=5 size 12{y+z=5} {}
(36)
z+w=3z+w=3 size 12{z+w=3} {}
(37)
x+w=5x+w=5 size 12{x+w=5} {}
(38)

Solution

(4, 3, 2, 1)

Exercise 24

x+w=6x+w=6 size 12{x+w=6} {}
(39)
2x+y+w=162x+y+w=16 size 12{2x+y+w="16"} {}
(40)
x−2z=0x−2z=0 size 12{x - 2z=0} {}
(41)
z+w=5z+w=5 size 12{z+w=5} {}
(42)

SYSTEMS OF LINEAR EQUATIONS – SPECIAL CASES

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

Exercise 25

2x+6y=82x+6y=8 size 12{2x+6y=8} {}
(43)
x+3y=4x+3y=4 size 12{x+3y=4} {}
(44)

Solution

(4−3t,t)(4−3t size 12{4 - 3t} {},t size 12{t} {})
(45)

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.

Exercise 27

2x−y=102x−y=10 size 12{2x - y="10"} {}
(46)
−4x+2y=15−4x+2y=15 size 12{ - 4x+2y="15"} {}
(47)

Solution

Inconsistent system, no solution

Exercise 28

x+y+z=6x+y+z=6 size 12{x+y+z=6} {}
(48)
3x+2y+z=143x+2y+z=14 size 12{3x+2y+z="14"} {}
(49)
4x+3y+2z=204x+3y+2z=20 size 12{4x+3y+2z="20"} {}
(50)

Exercise 29

x+2y−4z=1x+2y−4z=1 size 12{x+2y - 4z=1} {}
(51)
2x−3y+8z=92x−3y+8z=9 size 12{2x - 3y+8z=9} {}
(52)

Solution

(3−4/7t,−1+16/7t,t)(3−4/7t size 12{3 - 4/7t} {},−1+16/7t size 12{ - 1+"16"/7t} {},t size 12{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.

Exercise 31

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?

Solution

No, they are not consistent.

Exercise 32

x+y+2z=0x+y+2z=0 size 12{x+y+2z=0} {}
(53)
x+2y+z=0x+2y+z=0 size 12{x+2y+z=0} {}
(54)
2x+3y+3z=02x+3y+3z=0 size 12{2x+3y+3z=0} {}
(55)

Exercise 33

Find three solutions to the following system of equations.

x+2y+z=12x+2y+z=12 size 12{x+2y+z="12"} {}
(56)
y=3y=3 size 12{y=3} {}
(57)

Solution

(5, 3, 1), (4, 3, 2) (3, 3, 3)

Exercise 34

For what values of kk size 12{k} {} the following system of equations have a). No solution? b). Infinitely many solutions?

x+2y=5x+2y=5 size 12{x+2y=5} {}
(58)
2x+4y=k2x+4y=k size 12{2x+4y=k} {}
(59)

Exercise 35

x+3y−z=5x+3y−z=5 size 12{x+3y - z=5} {}
(60)

Solution

( 5−3s+t5−3s+t size 12{5 - 3s+t} {}, ss size 12{s} {}, tt size 12{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

73211−3−2773211−3−27 size 12{ left [ matrix { 7 {} # 3 {} ## 2 {} # 1{} } right ] left [ matrix { 1 {} # - 3 {} ## - 2 {} # 7{} } right ]} {}
(61)

Exercise 38

1−1010−123−43−412−413−511−1010−123−43−412−413−51 size 12{ left [ matrix { 1 {} # - 1 {} # 0 {} ## 1 {} # 0 {} # - 1 {} ## 2 {} # 3 {} # - 4{} } right ] left [ matrix { 3 {} # - 4 {} # 1 {} ## 2 {} # - 4 {} # 1 {} ## 3 {} # - 5 {} # 1{} } right ]} {}
(62)

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

Exercise 39

3−5−123−5−12 size 12{ left [ matrix { 3 {} # - 5 {} ## - 1 {} # 2{} } right ]} {}
(63)

Solution

25132513 size 12{ left [ matrix { 2 {} # 5 {} ## 1 {} # 3{} } right ]} {}
(64)

Exercise 40

102014001102014001 size 12{ left [ matrix { 1 {} # 0 {} # 2 {} ## 0 {} # 1 {} # 4 {} ## 0 {} # 0 {} # 1{} } right ]} {}
(65)

Exercise 41

11−110121111−1101211 size 12{ left [ matrix { 1 {} # 1 {} # - 1 {} ## 1 {} # 0 {} # 1 {} ## 2 {} # 1 {} # 1{} } right ]} {}
(66)

Solution

12–1–1–32–1–1112–1–1–32–1–11 size 12{ left [ matrix { 1 {} # 2 {} # –1 {} ## –1 {} # –3 {} # 2 {} ## –1 {} # –1 {} # 1{} } right ]} {}
(67)

Exercise 42

111310112111310112 size 12{ left [ matrix { 1 {} # 1 {} # 1 {} ## 3 {} # 1 {} # 0 {} ## 1 {} # 1 {} # 2{} } right ]} {}
(68)

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

Exercise 43

3x−5y=23x−5y=2 size 12{3x - 5y=2} {}
(69)
−x+2y=0−x+2y=0 size 12{ - x+2y=0} {}
(70)

Solution

(4, 2)

Exercise 44

x+2z=8y+4z=8z=3x+2z=8y+4z=8z=3 size 12{ matrix { x {} # +{} {} # {} # {} # 2z {} # ={} {} # 8 {} ## {} # {} # y {} # +{} {} # 4z {} # ={} {} # 8 {} ## {} # {} # {} # {} # z {} # ={} {} # 3{} } } {}
(71)

Exercise 45

x+y−z=2x+z=72x+y+z=13x+y−z=2x+z=72x+y+z=13 size 12{ matrix { x {} # +{} {} # y {} # - {} {} # z {} # ={} {} # 2 {} ## x {} # +{} {} # {} # {} # z {} # ={} {} # 7 {} ## 2x {} # +{} {} # y {} # +{} {} # z {} # ={} {} # "13"{} } } {}
(72)

Solution

(3, 3, 4)

Exercise 46

x+y+z=23x+y=7x+y+2z=3x+y+z=23x+y=7x+y+2z=3 size 12{ matrix { x {} # +{} {} # y {} # +{} {} # z {} # ={} {} # 2 {} ## 3x {} # +{} {} # y {} # {} # {} # ={} {} # 7 {} ## x {} # +{} {} # y {} # +{} {} # 2z {} # ={} {} # 3{} } } {}
(73)

Exercise 47

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.

Solution

If a matrix MM size 12{M} {} has an inverse, then the system of linear equations that has MM size 12{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 AX=BAX=B size 12{ ital "AX"=B} {} by the matrix inverse method, but discover AA size 12{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 2
A B C D E F G H I J K L M
1 2 3 4 5 6 7 8 9 10 11 12 13
N O P Q R S T U V W X Y Z
14 15 16 17 18 19 20 21 22 23 24 25 26

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

A = 3 2 1 1 A = 3 2 1 1 size 12{A= left [ matrix { 3 {} # 2 {} ## 1 {} # 1{} } right ]} {}

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.

Exercise 49

Encode the message: WATCH OUT!

Solution

712466237835873611447712466237835873611447 size 12{ left [ matrix { "71" {} ## "24" } right ] left [ matrix { "66" {} ## "23" } right ] left [ matrix { "78" {} ## "35" } right ] left [ matrix { "87" {} ## "36" } right ] left [ matrix { "114" {} ## "47" } right ]} {}
(74)

Exercise 50

Encode the message: HELP IS ON THE WAY.

Exercise 51

Decode the following message:

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

Solution

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.

B = 1 0 0 2 1 2 1 0 − 1 B = 1 0 0 2 1 2 1 0 − 1 size 12{B= left [ matrix { 1 {} # 0 {} # 0 {} ## 2 {} # 1 {} # 2 {} ## 1 {} # 0 {} # - 1{} } right ]} {}

In the two problems following, decode the messages that were encoded using matrix BB size 12{B} {}.

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

Exercise 53

Encode the message using matrix BB size 12{B} {}:

LUCK IS ON YOUR SIDE.

Solution

125191167219951414105−111587−3279118467−23125191167219951414105−111587−3279118467−23 size 12{ left [ matrix { "12" {} ## "51" {} ## 9 } right ] left [ matrix { "11" {} ## "67" {} ## 2 } right ] left [ matrix { "19" {} ## "95" {} ## "14" } right ] left [ matrix { "14" {} ## "105" {} ## - "11" } right ] left [ matrix { "15" {} ## "87" {} ## - 3 } right ] left [ matrix { "27" {} ## "91" {} ## "18" } right ] left [ matrix { 4 {} ## "67" {} ## - "23" } right ]} {}
(75)

Exercise 54

Encode the message using matrix BB size 12{B} {}:

MAY THE FORCE BE WITH YOU.

Exercise 55

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

Solution

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

Exercise 57

Solve the following homogeneous system.

x+y+z=0x+y+z=0 size 12{x+y+z=0} {}
(76)
3x+2y+z=03x+2y+z=0 size 12{3x+2y+z=0} {}
(77)
4x+3y+2z=04x+3y+2z=0 size 12{4x+3y+2z=0} {}
(78)

Solution

( tt size 12{t} {}, −2t−2t size 12{ - 2t} {}, tt size 12{t} {})

Exercise 58

Solve the following homogeneous system.

x−y−z=0x−y−z=0 size 12{x - y - z=0} {}
(79)
x−3y+2z=0x−3y+2z=0 size 12{x - 3y+2z=0} {}
(80)
2x−4y+z=02x−4y+z=0 size 12{2x - 4y+z=0} {}
(81)

Exercise 59

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?

Solution

Chris=$1250Chris=$1250 size 12{"Chris"=$"1250"} {}, Ed=$1,000Ed=$1,000 size 12{"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?

Exercise 61

Given the internal consumption matrix AA size 12{A} {}, and the external demand matrix DD size 12{D} {} as follows.

A=.30.20.10.20.10.30.10.20.30D=100150200A=.30.20.10.20.10.30.10.20.30 size 12{A= left [ matrix { "." "30" {} # "." "20" {} # "." "10" {} ## "." "20" {} # "." "10" {} # "." "30" {} ## "." "10" {} # "." "20" {} # "." "30"{} } right ]} {}D=100150200 size 12{D= left [ matrix { "100" {} ## "150" {} ## "200" } right ]} {}
(82)

Solve the system using the open model: X=AX+DX=AX+D size 12{X= ital "AX"+D} {} or X=I−A−1DX=I−A−1D size 12{X= left (I - A right ) rSup { size 8{ - 1} } D} {}

Solution

315.34383.52440.34315.34383.52440.34 size 12{ left [ matrix { "315" "." "34" {} ## "383" "." "52" {} ## "440" "." "34" } right ]} {}
(83)

Exercise 62

Given the internal consumption matrix AA size 12{A} {}, and the external demand matrix DD size 12{D} {} as follows.

A=.05.10.10.10.15.05.05.20.20D=5010080A=.05.10.10.10.15.05.05.20.20 size 12{A= left [ matrix { "." "05" {} # "." "10" {} # "." "10" {} ## "." "10" {} # "." "15" {} # "." "05" {} ## "." "05" {} # "." "20" {} # "." "20"{} } right ]} {}D=5010080 size 12{D= left [ matrix { "50" {} ## "100" {} ## "80" } right ]} {}
(84)

Solve the system using the open model: X=AX+DX=AX+D size 12{X= ital "AX"+D} {} or X=I−A−1DX=I−A−1D size 12{X= left (I - A right ) rSup { size 8{ - 1} } D} {}

Exercise 63

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?

Solution

Farming=$201,754.38,Building=$307,017.54Farming=$201,754.38 size 12{"Farming"=$"201","754" "." "38"} {},Building=$307,017.54 size 12{"Building"=$"307","017" "." "54"} {}
(85)

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?

Exercise 65

Suppose an economy consists of three industries FF size 12{F} {}, CC size 12{C} {}, and TT size 12{T} {}. The following table gives information about the internal use of each industry's production and external demand in dollars.

Table 3
  F F size 12{F} {} C C size 12{C} {} T T size 12{T} {} Demand Total
F F size 12{F} {} 30 10 20 40 100
C C size 12{C} {} 20 30 20 50 120
T T size 12{T} {} 10 10 30 60 110

Find the proportion of the amounts consumed by each of the industries; that is, find the matrix AA size 12{A} {}.

Solution

30/10010/12020/11020/10030/12020/11010/10010/12030/11030/10010/12020/11020/10030/12020/11010/10010/12030/110 size 12{ left [ matrix { "30"/"100" {} # "10"/"120" {} # "20"/"110" {} ## "20"/"100" {} # "30"/"120" {} # "20"/"110" {} ## "10"/"100" {} # "10"/"120" {} # "30"/"110"{} } right ]} {}
(86)

Exercise 66

If in the preceding problem, the consumer demand for FF size 12{F} {}, CC size 12{C} {}, and TT size 12{T} {} becomes 60, 80, and 100, respectively, find the total output and the internal use by each industry to meet that demand.

CHAPTER REVIEW

Exercise 67

To reinforce her diet, Mrs. Tam bought a bottle containing 30 tablets of Supplement AA size 12{A} {} and a bottle containing 50 tablets of Supplement BB size 12{B} {}. Each tablet of supplement AA size 12{A} {} contains 1000 mg of calcium, 400 mg of magnesium, and 15 mg of zinc, and each tablet of supplement BB size 12{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 2×32×3 size 12{2 times 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.

Solution

  1. 1000 400 15 800 500 20 1000 400 15 800 500 20 size 12{ left [ matrix { "1000" {} # "400" {} # "15" {} ## "800" {} # "500" {} # "20"{} } right ]} {}
  2. 30 50 30 50 size 12{ left [ matrix { "30" {} # "50"{} } right ]} {}

Exercise 68

Let matrix A=1−133−21A=1−133−21 size 12{A= left [ matrix { 1 {} # - 1 {} # 3 {} ## 3 {} # - 2 {} # 1{} } right ]} {} and B=33−114−3B=33−114−3 size 12{B= left [ matrix { 3 {} # 3 {} # - 1 {} ## 1 {} # 4 {} # - 3{} } right ]} {} . Find the following.

  1. 12A+B12A+B size 12{ { {1} over {2} } left (A+B right )} {}

  2. 3A−2B3A−2B size 12{3A - 2B} {}

Solution

  1. 2 1 1 2 1 − 1 2 1 1 2 1 − 1 size 12{ left [ matrix { 2 {} # 1 {} # 1 {} ## 2 {} # 1 {} # - 1{} } right ]} {}
  2. − 3 − 9 11 7 − 14 9 − 3 − 9 11 7 − 14 9 size 12{ left [ matrix { - 3 {} # - 9 {} # "11" {} ## 7 {} # - "14" {} # 9{} } right ]} {}
    (87)

Exercise 69

Let matrix C=11−1211101C=11−1211101 size 12{C= left [ matrix { 1 {} # 1 {} # - 1 {} ## 2 {} # 1 {} # 1 {} ## 1 {} # 0 {} # 1{} } right ]} {} and D=2−3−13−1−23−3−2D=2−3−13−1−23−3−2 size 12{D= left [ matrix { 2 {} # - 3 {} # - 1 {} ## 3 {} # - 1 {} # - 2 {} ## 3 {} # - 3 {} # - 2{} } right ]} {}. Find the following.

  1. 2C−D2C−D size 12{2 left (C - D right )} {}

  2. C−3DC−3D size 12{C - 3D} {}

Solution

  1. − 2 8 0 − 2 4 6 − 4 6 6 − 2 8 0 − 2 4 6 − 4 6 6 size 12{ left [ matrix { - 2 {} # 8 {} # 0 {} ## - 2 {} # 4 {} # 6 {} ## - 4 {} # 6 {} # 6{} } right ]} {}
  2. − 5 10 2 − 7 4 7 − 8 9 7 − 5 10 2 − 7 4 7 − 8 9 7 size 12{ left [ matrix { - 5 {} # "10" {} # 2 {} ## - 7 {} # 4 {} # 7 {} ## - 8 {} # 9 {} # 7{} } right ]} {}
    (88)

Exercise 70

Let matrix E=1−12312E=1−12312 size 12{E= left [ matrix { 1 {} # - 1 {} ## 2 {} # 3 {} ## 1 {} # 2{} } right ]} {} and F=21−112−3F=21−112−3 size 12{F= left [ matrix { 2 {} # 1 {} # - 1 {} ## 1 {} # 2 {} # - 3{} } right ]} {}. Find the following.

  1. 2EF2EF size 12{2 ital "EF"} {}

  2. 3FE3FE size 12{3 ital "FE"} {}

Solution

  1. 2 − 2 4 14 16 − 22 8 10 − 14 2 − 2 4 14 16 − 22 8 10 − 14 size 12{ left [ matrix { 2 {} # - 2 {} # 4 {} ## "14" {} # "16" {} # - "22" {} ## 8 {} # "10" {} # - "14"{} } right ]} {}
  2. 9 − 3 6 − 3 9 − 3 6 − 3 size 12{ left [ matrix { 9 {} # - 3 {} ## 6 {} # - 3{} } right ]} {}
    (89)

Exercise 71

Let matrix G=1−13321G=1−13321 size 12{G= left [ matrix { 1 {} # - 1 {} # 3 {} ## 3 {} # 2 {} # 1{} } right ]} {} and H=abcdefH=abcdef size 12{H= left [ matrix { a {} # b {} ## c {} # d {} ## e {} # f{} } right ]} {} . Find the following.

  1. 2GH2GH size 12{2 ital "GH"} {}

  2. HGHG size 12{ ital "HG"} {}

Solution

  1. 2a − 2c + 6 e 2b − 2d + 6f 6a + 4c + 2 e 6b + 4d + 2f 2a − 2c + 6 e 2b − 2d + 6f 6a + 4c + 2 e 6b + 4d + 2f size 12{ left [ matrix { 2a - 2c+6e {} # 2b - 2d+6f {} ## 6a+4c+2e {} # 6b+4d+2f{} } right ]} {}
  2. a + 3b − a + 2b 3a + b c + 3d − c + 2d 3c + d e + 3f − e + 2f 3e + f a + 3b − a + 2b 3a + b c + 3d − c + 2d 3c + d e + 3f − e + 2f 3e + f size 12{ left [ matrix { a+3b {} # - a+2b {} # 3a+b {} ## c+3d {} # - c+2d {} # 3c+d {} ## e+3f {} # - e+2f {} # 3e+f{} } right ]} {}
    (90)

Exercise 72

Solve the following systems using the Gauss-Jordan Method.

  1. x+3y−2z=72x+7y−5z=1x+5y−3z=1x+3y−2z=72x+7y−5z=1x+5y−3z=1 size 12{ matrix { x {} # +{} {} # 3y {} # - {} {} # 2z {} # ={} {} # 7 {} ## 2x {} # +{} {} # 7y {} # - {} {} # 5z {} # ={} {} # 1 {} ## x {} # +{} {} # 5y {} # - {} {} # 3z {} # ={} {} # 1{} } } {}

  2. 2x−4y+4z=22x+y+9z=13x−2y+2z=72x−4y+4z=22x+y+9z=13x−2y+2z=7 size 12{ matrix { 2x {} # - {} {} # 4y {} # +{} {} # 4z {} # ={} {} # 2 {} ## 2x {} # +{} {} # y {} # +{} {} # 9z {} # ={} {} # 1 {} ## 3x {} # - {} {} # 2y {} # +{} {} # 2z {} # ={} {} # 7{} } } {}

Solution

  1. (2, 1, –1)
  2. (3, 2, 1)

Exercise 73

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.

Solution

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

Exercise 74

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. x+y+z=62x−3y+2z=13x−2y+3z=1x+y+z=62x−3y+2z=13x−2y+3z=1 size 12{ matrix { x {} # +{} {} # y {} # +{} {} # z {} # ={} {} # 6 {} ## 2x {} # - {} {} # 3y {} # +{} {} # 2z {} # ={} {} # 1 {} ## 3x {} # - {} {} # 2y {} # +{} {} # 3z {} # ={} {} # 1{} } } {}

  2. x+y+3z=4x+z=12x−y=2x+y+3z=4x+z=12x−y=2 size 12{ matrix { x {} # +{} {} # y {} # +{} {} # 3z {} # ={} {} # 4 {} ## x {} # {} # {} # +{} {} # z {} # ={} {} # 1 {} ## 2x {} # - {} {} # y {} # {} # {} # ={} {} # 2{} } } {}

Solution

  1. x=6−tx=6−t size 12{x=6 - t} {}, y=0y=0 size 12{y=0} {}, z=tz=t size 12{z=t} {}; (5, 0, 1)

  2. no solution

Exercise 75

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.

Solution

n=3t−12n=3t−12 size 12{n=3t - "12"} {}, d=−4t+24d=−4t+24 size 12{d= - 4t+"24"} {}, q=tq=t size 12{q=t} {}; n=3n=3 size 12{n=3} {}, d=4d=4 size 12{d=4} {}, q=5q=5 size 12{q=5} {}

Exercise 76

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. x+2y=42x+4y=83x+6y−3z=3x+2y=42x+4y=83x+6y−3z=3 size 12{ matrix { x {} # +{} {} # 2y {} # {} # {} # ={} {} # 4 {} ## 2x {} # +{} {} # 4y {} # {} # {} # ={} {} # 8 {} ## 3x {} # +{} {} # 6y {} # - {} {} # 3z {} # ={} {} # 3{} } } {}

  2. x − 2y + 2z = 1 2x − 3y + 5z = 4 x − 2y + 2z = 1 2x − 3y + 5z = 4 size 12{ matrix { x - 2y+2z=1 {} ## 2x - 3y+5z=4 } } {}

Solution

  1. x=4−2tx=4−2t size 12{x=4 - 2t} {}, y=ty=t size 12{y=t} {}, z=3z=3 size 12{z=3} {}; (4, 0, 3)

  2. x=5−4tx=5−4t size 12{x=5 - 4t} {}, y=2−ty=2−t size 12{y=2 - t} {}, z=tz=t size 12{z=t} {}; (1, 1, 1)

Exercise 77

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. 2x+y−2z=02x+2y−3z=06x+4y−7z=02x+y−2z=02x+2y−3z=06x+4y−7z=0 size 12{ matrix { 2x {} # +{} {} # y {} # - {} {} # 2z {} # ={} {} # 0 {} ## 2x {} # +{} {} # 2y {} # - {} {} # 3z {} # ={} {} # 0 {} ## 6x {} # +{} {} # 4y {} # - {} {} # 7z {} # ={} {} # 0{} } } {}

  2. 3x+4y−3z=52x+3y−z=4x+2y+z=13x+4y−3z=52x+3y−z=4x+2y+z=1 size 12{ matrix { 3x {} # +{} {} # 4y {} # - {} {} # 3z {} # ={} {} # 5 {} ## 2x {} # +{} {} # 3y {} # - {} {} # z {} # ={} {} # 4 {} ## x {} # +{} {} # 2y {} # +{} {} # z {} # ={} {} # 1{} } } {}

Solution

  1. x=.5tx=.5t size 12{x= "." 5t} {}, y=ty=t size 12{y=t} {}, z−2tz−2t size 12{z - 2t} {}; (1, 2, 2)

  2. no solution

Exercise 78

Find the inverse of the following matrices.

  1. 23352335 size 12{ left [ matrix { 2 {} # 3 {} ## 3 {} # 5{} } right ]} {}

  2. 111121231111121231 size 12{ left [ matrix { 1 {} # 1 {} # 1 {} ## 1 {} # 2 {} # 1 {} ## 2 {} # 3 {} # 1{} } right ]} {}

Solution

  1. 5 − 3 − 3 2 5 − 3 − 3 2 size 12{ left [ matrix { 5 {} # - 3 {} ## - 3 {} # 2{} } right ]} {}
    (91)
  2. 1 − 2 1 − 1 1 0 1 1 − 1 1 − 2 1 − 1 1 0 1 1 − 1 size 12{ left [ matrix { 1 {} # - 2 {} # 1 {} ## - 1 {} # 1 {} # 0 {} ## 1 {} # 1 {} # - 1{} } right ]} {}
    (92)

Exercise 79

Solve the following systems using the matrix inverse method.

  1. 2x+3y+z=1x+2y+z=9x+y+z=52x+3y+z=1x+2y+z=9x+y+z=5 size 12{ matrix { 2x {} # +{} {} # 3y {} # + {} {} # z {} # ={} {} # 1 {} ## x {} # +{} {} # 2y {} # +{} {} # z {} # ={} {} # 9 {} ## x {} # +{} {} # y {} # +{} {} # z {} # ={} {} # 5{} } } {}

  2. x+2y−3z+w=7x−z=4x−2y+z=0y−2z+w=-x+2y−3z+w=7x−z=4x−2y+z=0y−2z+w=- size 12{ matrix { x {} # +{} {} # 2y {} # - {} {} # 3z {} # +{} {} # w {} # ={} {} # 7 {} ## x {} # {} # {} # - {} {} # z {} # {} # {} # ={} {} # 4 {} ## x {} # - {} {} # 2y {} # +{} {} # z {} # {} # {} # ={} {} # 0 {} ## {} # {} # y {} # - {} {} # 2z {} # +{} {} # w {} # ={} {} # _{} } } {}

Solution

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

Exercise 80

Use matrix AA size 12{A} {}, given below, to encode the following messages. The space between the letters is represented by the number 27, and all punctuation is ignored.

A=120121010A=120121010 size 12{A= left [ matrix { 1 {} # 2 {} # 0 {} ## 1 {} # 2 {} # 1 {} ## 0 {} # 1 {} # 0{} } right ]} {}
(93)
  1. TAKE IT AND RUN.
  2. GET OUT QUICK.

Solution

  1. 22 33 1 59 68 27 74 75 27 22 49 4 60 74 21 22 33 1 59 68 27 74 75 27 22 49 4 60 74 21 size 12{ left [ matrix { "22" {} ## "33" {} ## 1 } right ] left [ matrix { "59" {} ## "68" {} ## "27" } right ] left [ matrix { "74" {} ## "75" {} ## "27" } right ] left [ matrix { "22" {} ## "49" {} ## 4 } right ] left [ matrix { "60" {} ## "74" {} ## "21" } right ]} {}
    (94)
  2. 17 37 5 57 78 15 74 91 27 39 42 9 65 92 27 17 37 5 57 78 15 74 91 27 39 42 9 65 92 27 size 12{ left [ matrix { "17" {} ## "37" {} ## 5 } right ] left [ matrix { "57" {} ## "78" {} ## "15" } right ] left [ matrix { "74" {} ## "91" {} ## "27" } right ] left [ matrix { "39" {} ## "42" {} ## 9 } right ] left [ matrix { "65" {} ## "92" {} ## "27" } right ]} {}
    (95)

Exercise 81

Decode the following messages that were encoded using matrix AA size 12{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

Solution

  1. NO PAIN NO GAIN
  2. GO FOR THE GOLD

Exercise 82

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?

Solution

x=40/33t, y=36/33t, z=t; Chris=40hrs, Bob=36hrs, Matt=33hrsx=40/33t size 12{x="40"/"33"t} {}, y=36/33t size 12{y="36"/"33"t} {}, z=t size 12{z=t} {}; Chris=40hrs size 12{"Chris"="40""hrs"} {}, Bob=36hrs size 12{"Bob"="36hrs"} {}, Matt=33hrs size 12{"Matt"="33hrs"} {}
(96)

Exercise 83

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?

Solution

Chris=34.1hrs, Bob=32.2hrs, Matt=35.2hrsChris=34.1hrs size 12{"Chris"="34" "." "1hrs"} {}, Bob=32.2hrs size 12{"Bob"="32" "." "2hrs"} {}, Matt=35.2hrs size 12{"Matt"="35" "." "2hrs"} {}
(97)

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