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Inside Collection:

Collection by: Rupinder Sekhon. E-mail the author

# Matrices: Homework

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

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, JFJF 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

3A2B3A2B size 12{3A - 2B} {}
(3)

#### Solution

7201255010172012550101 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

4x5y=64x5y=6 size 12{4x - 5y=6} {}
(13)
5x6y=75x6y=7 size 12{5x - 6y=7} {}
(14)

#### Solution

4556xy=674556xy=67 size 12{ left [ matrix { 4 {} # - 5 {} ## 5 {} # - 6{} } right ] left [ matrix { x {} ## y } right ]= left [ matrix { 6 {} ## 7 } right ]} {}
(15)

### Exercise 14

x2y+2z=3x2y+2z=3 size 12{x - 2y+2z=3} {}
(16)
x3y+4z=7x3y+4z=7 size 12{x - 3y+4z=7} {}
(17)
x2y3z=12x2y3z=12 size 12{x - 2y - 3z= - "12"} {}
(18)

### Exercise 15

2x+3z=172x+3z=17 size 12{2x+3z="17"} {}
(19)
3x2y=103x2y=10 size 12{3x - 2y="10"} {}
(20)
5y+2z=115y+2z=11 size 12{5y+2z="11"} {}
(21)

#### Solution

203320052xyz=171011203320052xyz=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=14x2yz=5y2z+4w=9x+3z+3w=15x+2y+3z+2w=14x2yz=5y2z+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)
2x5y=132x5y=13 size 12{2x - 5y="13"} {}
(25)

(4,-1)

### Exercise 18

xyz=1xyz=1 size 12{x - y - z= - 1} {}
(26)
x3y+2z=7x3y+2z=7 size 12{x - 3y+2z=7} {}
(27)
2xy+z=32xy+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)
2xy+2z=112xy+2z=11 size 12{2x - y+2z="11"} {}
(31)

(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.

(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)

(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)
x2z=0x2z=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

(43t,t)(43t 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

2xy=102xy=10 size 12{2x - y="10"} {}
(46)
4x+2y=154x+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+2y4z=1x+2y4z=1 size 12{x+2y - 4z=1} {}
(51)
2x3y+8z=92x3y+8z=9 size 12{2x - 3y+8z=9} {}
(52)

#### Solution

(34/7t,1+16/7t,t)(34/7t size 12{3 - 4/7t} {},1+16/7t size 12{ - 1+"16"/7t} {},t size 12{t} {})

### Exercise 30

#### Solution

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

### 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.

 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=113321A=113321 size 12{A= left [ matrix { 1 {} # - 1 {} # 3 {} ## 3 {} # - 2 {} # 1{} } right ]} {} and B=331143B=331143 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. 3A2B3A2B 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=111211101C=111211101 size 12{C= left [ matrix { 1 {} # 1 {} # - 1 {} ## 2 {} # 1 {} # 1 {} ## 1 {} # 0 {} # 1{} } right ]} {} and D=231312332D=231312332 size 12{D= left [ matrix { 2 {} # - 3 {} # - 1 {} ## 3 {} # - 1 {} # - 2 {} ## 3 {} # - 3 {} # - 2{} } right ]} {}. Find the following.

1. 2CD2CD size 12{2 left (C - D right )} {}

2. C3DC3D 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=112312E=112312 size 12{E= left [ matrix { 1 {} # - 1 {} ## 2 {} # 3 {} ## 1 {} # 2{} } right ]} {} and F=211123F=211123 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=113321G=113321 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+3y2z=72x+7y5z=1x+5y3z=1x+3y2z=72x+7y5z=1x+5y3z=1 size 12{ matrix { x {} # +{} {} # 3y {} # - {} {} # 2z {} # ={} {} # 7 {} ## 2x {} # +{} {} # 7y {} # - {} {} # 5z {} # ={} {} # 1 {} ## x {} # +{} {} # 5y {} # - {} {} # 3z {} # ={} {} # 1{} } } {}

2. 2x4y+4z=22x+y+9z=13x2y+2z=72x4y+4z=22x+y+9z=13x2y+2z=7 size 12{ matrix { 2x {} # - {} {} # 4y {} # +{} {} # 4z {} # ={} {} # 2 {} ## 2x {} # +{} {} # y {} # +{} {} # 9z {} # ={} {} # 1 {} ## 3x {} # - {} {} # 2y {} # +{} {} # 2z {} # ={} {} # 7{} } } {}

1. (2, 1, –1)
2. (3, 2, 1)

### 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=62x3y+2z=13x2y+3z=1x+y+z=62x3y+2z=13x2y+3z=1 size 12{ matrix { x {} # +{} {} # y {} # +{} {} # z {} # ={} {} # 6 {} ## 2x {} # - {} {} # 3y {} # +{} {} # 2z {} # ={} {} # 1 {} ## 3x {} # - {} {} # 2y {} # +{} {} # 3z {} # ={} {} # 1{} } } {}

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

#### Solution

1. x=6tx=6t 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=3t12n=3t12 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+6y3z=3x+2y=42x+4y=83x+6y3z=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=42tx=42t size 12{x=4 - 2t} {}, y=ty=t size 12{y=t} {}, z=3z=3 size 12{z=3} {}; (4, 0, 3)

2. x=54tx=54t size 12{x=5 - 4t} {}, y=2ty=2t 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+y2z=02x+2y3z=06x+4y7z=02x+y2z=02x+2y3z=06x+4y7z=0 size 12{ matrix { 2x {} # +{} {} # y {} # - {} {} # 2z {} # ={} {} # 0 {} ## 2x {} # +{} {} # 2y {} # - {} {} # 3z {} # ={} {} # 0 {} ## 6x {} # +{} {} # 4y {} # - {} {} # 7z {} # ={} {} # 0{} } } {}

2. 3x+4y3z=52x+3yz=4x+2y+z=13x+4y3z=52x+3yz=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} {}, z2tz2t 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+2y3z+w=7xz=4x2y+z=0y2z+w=-x+2y3z+w=7xz=4x2y+z=0y2z+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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| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

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##### What are tags?

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