# Matrices and Linear Programming

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Solve the problem.

1) An appliance store sells two types of refrigerators. Each Cool-It refrigerator sells for $ 640 and each Polar sells for $ 740. Up to 330 refrigerators can be stored in the warehouse and new refrigerators are delivered only once a month. It is known that customers will buy at least 80 Cool-Its and at least 100 Polars each month. How many of each brand should the store stock and sell each month to maximize revenues? 1) _______

A) 230 Cool-Its and 100 Polars B) 310 Cool-Its and 175 Polars

C) 80 Cool-Its and 250 Polars D) 95 Cool-Its and 235 Polars

Provide an appropriate response.

2) Give the dimensions of the following matrix.

| 1 3 -1 2 4|

| 3 5 1 0 6 | 2) _______

A) 5 x 2 B) 2 x 2 C) 2 x 5 D) 10 x 1

Use the Gauss-Jordan method to solve the system of equations.

3) 3x + 3y = -6

2x + 8y = 14 3) _______

A) ( 3, -5) B) ( -5, 3) C) ( -5, -3) D) No solution

Each day Larry needs at least 10 units of vitamin A, 12 units of vitamin B, and 20 units of vitamin C. Pill #1 contains 4 units of A and 3 of B. Pill #2 contains 1 unit of A, 2 of B, and 4 of C. Pill #3 contains 10 units of A, 1 of B, and 5 of C.

4) Pill #1 costs 9 cents, pill #2 costs 8 cents, and pill #3 costs 10 cents. Larry wants to minimize cost. What are the coefficients of the objective function? 4) _______

A) 4, 1, 10 B) 9, 4, 3 C) 10, 12, 20 D) 9, 8, 10

A manufacturing company wants to maximize profits on products A, B, and C. The profit margin is $3 for A, $6 for B, and $15 for C. The production requirements and departmental capacities are as follows:

Department Production requirement

by product (hours) Departmental capacity

(Total hours)

A B C

Assembling 2 3 2 30,000

Painting 1 2 2 38,000

Finishing 2 3 1 28,000

5) What are the coefficients of the objective function? 5) _______

A) 1, 2, 2 B) 2, 3, 2 C) 2, 3, 1 D) 3, 6, 15

Solve using artificial variables.

6) Maximize z = 3X1 + 2x2

subject to: X1 +x2= 5

4X1 +2x2 ≥ 12

5X1 + 2x2 ≤ 16

X1≥0 + x2≥0 6) _______

A) Maximum is 14 for X1=4, X2=1

B) Maximum is 12 for X1=2, X2=3

C) Maximum is 13 for X1=3, X2=2

D) Maximum is 15 for X1=5, X2=0

Use slack variables to convert the constraints into linear equations.

7) Maximize z = 2X1 +8x2

subject to: X1 + 2x2 ≤ 15

8X1 + 2x2 ≤ 25

with: X1≥0 + x2≥0 7) _______

A) X1 + 2x2 + s1 = 15

8 X1 + 2x2 + s2 = 25

B) X1 + 2x2 = s1 + 15

8X1 + 2x2 = s2 + 25

C) X1 + 2x2 + s1 = 15

8X1 + 2x2 + s1 = 25

D) X1 + 2x2 + s1 ≤ 15

8X1 + 2x2 + s2 ≤ 25

A manufacturing company wants to maximize profits on products A, B, and C. The profit margin is $3 for A, $6 for B, and $15 for C. The production requirements and departmental capacities are as follows:

Department Production requirement

by product (hours) Departmental capacity

(Total hours)

A B C

Assembling 2 3 2 30,000

Painting 1 2 2 38,000

Finishing 2 3 1 28,000

8) What are the constants in the model? 8) _______

A) 3, 6, 15 B) 1, 2, 2

C) 2, 3, 3 D) 30,000, 38,000, 28,000

Rewrite the objective function into a maximization function.

9) Minimize w = 2y1 + 4y2 + 3y3

subject to: y1 + y2 ≥ 10

2y1 + 3y2 + y3 ≥ 27

y1 + 2 y2 + y3 ≥ 15

y1 ≥ 0, y2≥0, y3≥0 9) _______

A) Maximize z = -2x1 - 4x2 - 3x3

B) Maximize z = 2x1 + 4x2 - 3x3

C) Maximize z = -x1 - x2 ≤ 10

D) Maximize z = -2x1 -3 x2 - x3 ≤ 27

Solve the problem.

10) A company makes three chocolate candies: cherry, almond, and raisin. Matrix A gives the amount of ingredients in one batch. Matrix B gives the costs of ingredients from suppliers X and Y. Multiply the matrices.

Sugar Choc Milk

4 6 1 Cherry

A= 5 3 1 Almond

3 3 1 raisin

X Y

3 2 Sugar

B= 3 4 Choc

2 2 milk

10) ______

A)

X Y

32 34 Sugar

26 24 Choc

20 20 milk

B)

X Y

33 22 Cherry

27 36 Almond

14 14 Raisin

C)

X Y

22 33 Cherry

36 27 Almond

14 14 Raisin

D)

X Y

32 34 Cherry

26 24 Almond

20 20 Raisin

The initial tableau of a linear programming problem is given. Use the simplex method to solve the problem.

11)

11) ______

X1 X2 X3 S1 S2 z

3 2 4 1 0 0 18

2 1 5 0 1 0 8

-1 -4 -2 0 0 1 0

A) Maximum at 32 for x2=8, s1=2

B) Maximum at 36 for x2=2, s1=8

C) Maximum at 18 for x2=8, x3=2

D) Maximum at 9 for x1=8, x2=2

Write the solutions that can be read from the simplex tableau.

12)

12) ______

X1 X2 X3 S1 S2 z

3 4 0 3 1 0 12

1 5 1 7 0 0 23

-3 4 0 1 0 1 19

A) x1,x2, s1=0, x1=23, s2=12,z=19

B) x1,x2, s1=0, x3=23, s2=12,z=19

C) x1,x2, s1=0, x5=23, s2=12,z=19

D) x1,x2, s1=0, x3=12, s2=23,z=19

Perform the indicated operation where possible.

13)

-1 0 -1 3

3 3 - 3 1

13) ______

A)

0 3

0 -2

B) | -1 |

C)

-2 3

6 4

D)

0 -3

0 2

Solve the problem.

14) Factories A and B sent rice to stores 1 and 2. A sent 14 loads and B sent 21. Store 1 received 20 loads and store 2 received 15. It cost $200 to ship from A to 1, $350 from A to 2, $300 from B to 1, and $250 from B to 2. $ 8350 was spent. How many loads went where? 14) ______

A) 14 from A to 1, 0 from A to 2, 6 from B to 1, 15 from B to 2

B) 13 from A to 1, 1 from A to 2, 7 from B to 1, 4 from B to 2

C) 12 from A to 1, 2 from A to 2, 8 from B to 1, 13 from B to 2

D) 0 from A to 1, 14 from A to 2, 15 from B to 1, 6 from B to 2

Find the values of the variables in the matrix.

15)

-2 5 x m 5 4

3 y -4 = n -8 p

15) ______

A) m = 3, x = 5, n = -2, y = -8, p = -4

B) m = -2, x = 4, n = 3, y = -8, p = -4

C) m = -2, x = 5, n = 3, y = -8, p = -4

D) m = -2, x = 4, n = 5, y = -8, p = -4

Solve the problem.

16)

Let A = -3 6 Find 4A.

0 2

16) ______

1 10 -12 24 -12 24 -12 6

A) 4 6 B) 0 2 C) 0 8 D) 0 2

Find the values of the variables in the matrix.

17) 7 -8 x y

8 -1 = 8 z

17) ______

A) x = 7, y = -8, z = 8 B) x = -8, y = 7, z = -1

C) x = 7, y = -8, z = -1 D) x = 7, y = 8, z = -1

Write a matrix to display the information.

18) Factories A and B sent rice to stores 1 and 2. It cost $200 to ship from A to 1, $350 from A to 2, $300 from B to 1, and $250 from B to 2. Make a 2 × 2 matrix showing the shipping costs. Assign the factories to the rows and the stores to the columns. 18) ______

300 250 200 300 350 250 200 350

A) 350 200 B) 350 250 C) 200 300 D) 300 250

Solve the problem.

19) Barges from ports X and Y went to cities A and B. X sent 32 barges and Y sent 8. City A received 22 barges and B received 18. Shipping costs $220 from X to A, $300 from X to B, $400 from Y to A, and $180 from Y to B. $ 9280 was spent. How many barges went where? 19) ______

A) 22 from X to A, 10 from X to B, 0 from Y to A, 8 from Y to B

B) 20 from X to A, 12 from X to B, 2 from Y to A, 6 from Y to B

C) 16 from X to A, 16 from X to B, 6 from Y to A, 2 from Y to B

D) 18 from X to A, 18 from X to B, 4 from Y to A, 4 from Y to B

Convert the inequality into a linear equation by adding a slack variable.

20) x1 + 8x2 ≤ 19 20) ______

A) x1 + 8x2 + s1 ≤ 19

B) x1 + 8x2 + s1 + 19 =0

C) x1 + 8x2 + s1 < 19

D) x1 + 8x2 + s1 = 19

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#### Solution Summary

Step by step solutions to all the problems are provided.

Linear Programming : Optimizing using Matrix Methods

Matrix methods can be used to solve linear programming problems. A linear programming problem is used to find an optimal solution, subject to stated restraints.

For example, consider an accountant who prepares tax returns. Suppose a form 1040EZ requires $12 in computer resources to process and 22 minutes of the accountant's time. Assume a form 1040A takes $25 in computer resources and needs 48 minutes of the accountant's time. If the accountant can spend $630 on computer resources and has 1194 minutes available, how many forms of 1040EZ and 1040A can the accountant process?

2. You are given the following system of linear equations:

x - y + 2z = 13

2x + y - z = -6

-x + 3y + z = -7

a. Provide a coefficient matrix corresponding to the system of linear equations.

b. What is the inverse of this matrix?

c. What is the transpose of this matrix?

d. Find the determinant for this matrix.

3. Calculate the following for

a. A * B

b. -4A

c. AT

4. Solve the following linear system using Gaussian elimination.

Show work.

3x + y - z = -5

-4x + y = 6

6x - 2y + 3z = 2

5. Solve the following linear system for x using Cramer's rule.

Show work.

x + 2y - 3z = -22

2x - 6y + 8z = 74

-x - 2y + 4z = 29

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