# Matrices and Linear Programming

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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

#### Solution Summary

Step by step solutions to all the problems are provided.