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MATH3602: B. Sc. Operations Research

Tutorial Questions 5 Dr. John O. Mubenwafor

Matrices

5.1.1: Simple Operations With Matrices

1. Let [pic], [pic], [pic], [pic], [pic], F = [6 2], [pic], [pic], J = [4].

(a) State the order of each matrix.

(b) Which matrices are square?

(c) Which matrices are upper triangular? lower triangular?

(d) Which are row vectors?

(e) Which are column vectors?

[Answer: (a) 2X3, 3X3, 3X2, 2X2, 4X4, 1X2, 3X1, 3X3, 1X1

(b) B, D, E, H, J

(c) H, J upper triangular, D, J lower triangular

(d) F, J

(e) G, J]

2. Let [pic].

(a) What is the order of A?

(b) Find the following entries: [pic], [pic], [pic], [pic], [pic], [pic].

(c) What are the main-diagonal entries?

[Answer: (a) 4X4 (b) 2, –2, 4, 6 (c) 7, 2, 1, 0]

5.1.2: The transpose of a Matrix

1. Determine the transpose of the following:

(a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

[Answer: (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]]

2. Determine the transpose of: [pic] CCCJ 12/2009

[Answer: [pic]]

3. If [pic], find BT. [Answer: [pic]]

5.2: Matrix addition and subtraction

1. Given the following matrices:

[pic] [pic] [pic]

Calculate:

a) A + C (b) A – C (c) A + B and A – B

[Answer: (a) [pic] (b) [pic] (c) Not possible, no corresponding elements]

2. If [pic] and [pic], find A + B.

[Answer: [pic]]

3. Let [pic], [pic],

[pic], [pic].

a) Show that A + B = B + A.

b) Show that A + (B + C) = (A + B) + C.

c) Show that A + O = A

[Answer: (a) [pic] (b) [pic] (c) [pic]]

4. If [pic], and [pic], find: (i) A + B (ii) A – B

[Answer: (i) [pic], (ii) [pic]]

5.3.1: Scalar Multiplication

1. Let: [pic], [pic], [pic], [pic]

Compute the following:

a) 4A (b) ⅔B (c) ½A + 3B

d) –C (e) kC (f) 5D

[Answer: (a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) [pic]]

2. Let [pic], [pic], [pic],

[pic], [pic].

(i) Find: A + B, (ii) 2A – 4B, (iii) (A + B)T

(iv) 2C – 3D (v) Are E + ET, B + E and AT + E defined?

(vi) Find A + AT, and A – AT and determine whether they are symmetric or skew-symmetric.

[Answer: (i) [pic], (ii) [pic], (iii) [pic], (iv) [pic]

(v) not defined (vi) 2A, 2I, A and AT are symmetric but A + AT and A – AT are neither symmetric nor skew-symmetric]

3. Determine the matrix –A if [pic] CCCJ 12/2009

[Answer: [pic]]

5.3.2: Matrix Multiplication

1. Given the matrices:

[pic], [pic], [pic]

Evaluate:

(a) AC (b) CA (c) AB (d) BA

[Answer: (a) [pic], (b) [pic]]

2. Determine the matrix AB if [pic] and [pic]

[Answer: 10] CCCJ 12/2009

3. Given the matrices:

[pic], [pic] [pic]

Calculate, if possible:

a) A + B (f) A + C (k) (AB)T

b) A – B (g) A + BT (l) C + 5I

c) A + 4B (h) BC (m) CTA

d) A + I (i) CB (n) (BC)T

e) AI (j) CBT (o) AC + B

[Answer: (a) [pic] (b) [pic] (c) [pic] (d) [pic]

(e) [pic] (f) not possible (g) [pic] (h) [pic]

(i) not possible (j) not possible (k) [pic] (l) not possible

(m) [pic] (n) [pic] (o) Not possible]

4. If [pic], and [pic],

i) determine AB;

ii) Is AB the same as BA?

[Answer: (i) [pic], (ii) [pic], [pic]AB ≠ BA]

5. If [pic], find C2. [Answer: [pic]]

6. If [pic], [pic], [pic], and [pic],

find (a) AB (b) CD.

[Answer: (a) [pic] (b) [pic]]

7. A fast-food chain has three shops, A, B and C. The average daily sales and profit in each shop is given in the following tables:

| | |

|Units sold |Units profit |

|Shop A

|Shop B

|Shop C

|Shop A

|Shop B |Shop C | |Burgers |800 |400 |500 |20¢ |40¢ |33¢ | |Chips |950 |600 |700 |50¢ |45¢ |60¢ | |Drinks |500 |1200 |900 |30¢ |35¢ |20¢ | |

Use matrix multiplication to determine:

a) The profit for each product

b) The profit for each shop

[Answer: (a) burgers = $485, chips = $1165, drinks = $750

(b) shop A = $785, shop B = $850, shop C = $765]

5.4 Determinants

1. Evaluate the following determinants:

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

[Answer: (a) 7 (b) –70 (c) 27 (d) 0 (e) 126 (f) 24]

2. If [pic] and [pic], find: (i) det A (ii) det B and show that det (AB) = det (BA)

[Answer: (i) – 3 (ii) 1, – 3]

3. If [pic] and [pic], find: (i) AB (ii) BA (iii) Hence, or otherwise, show that det (AB) = det (BA)

[Answer: (i) [pic] (ii) [pic], (iii) – 40 ]

4. If [pic] and [pic],

find (i) det A,

(ii) det B,

(iii) det (2A)

(iv) det (2B)

[Answer: (i) 52 (ii) 72 (iii) 416 (iv) 576]

5.5 Minors, Cofactors, Adjoints & Inverse

1. If [pic],

(i) find [pic], [pic], [pic];

(ii) verify that the value of det A is: [pic].

[Answer: (i) – 4, 13, 6; (ii) 92]

2. Find the inverse of the following matrices:

(a) [pic] (b) [pic]

(c) [pic] (d) [pic].

[Answer: (a) [pic] (b) [pic]

(c) [pic] (d) [pic]]

3. Find the inverse of [pic]. [Answer: [pic]]

4. If [pic],

(i) Compute the determinant of A;

(ii) Find adjoint A;

(iii) Verify that A(adjoint A) = (det A)In;

(iv) Hence, find A-1.

[Answer: (i) 7 (ii) [pic], (iii) [pic], (iv) [pic]]

5. If [pic], [pic], and [pic]

(a) Find A-1, B-1, and B-1A-1;

(b) Hence, prove that: CB-1A-1 = In.

[Answer: (a) [pic], [pic], [pic] (b) [pic]]

6. If [pic],

(i) Compute the Ad joint A

(ii) Find the determinant of A

(iii) Hence, or otherwise, calculate the inverse (A-1) of A.

[Answer: (i) [pic] (ii) -200 (iii) [pic]]

7. If [pic], and f(x) = x2 – 4x + 3, find A2 and f(A).

[Answer: [pic], [pic]]

8. If [pic], and f(x) = x3 – 3x2 – 2x + 4, find A2, A3 and f(A).

[Answer: [pic], [pic], [pic]]

5.6 Systems of Equations: Solution by Inverse Matrix & Cramer’s Rule

1. Solve the following linear systems by any Inverse method:

(a) [pic] (b) [pic] (c) [pic]

[Answer: (a) 1, 4 (b) 8.8, – 3.3 (c) 2, 1]

2. Solve the following linear systems by any matrix method:

(a) [pic] (b) [pic]

(c) [pic] (d) [pic]

[Answer: (a) 2, 1, 2 (b) 4, 1, 0 (c) ½, 2, 3/2 (d) 1, 2, 3]

3. Solve the following systems of equations using inverse matrix method:

(a) [pic] (b) [pic]

(c) [pic]

[Answer: (a) 1, – 1, 2 (b) 1, 2, 3 (c) 3, -1, 2]

4. A supervisor and 7 workers together earn $520 per week, whilst 2 supervisors and 17 workers together earn $1220 per week. Using inverse method, calculate the weekly wage of a supervisor and of a worker.

[Answer: $100, $60]

5 When petrol cost 10¢ a litre and oil 25¢ a litre, a motorist found that his cost in petrol and oil was $10.50 for each 1000km. When petrol was increased to 10½¢ a litre and oil to 26¢ a litre, the cost was $11.02. Find how many litres of petrol and oil he used to travel 1000km.

[Answer: 100 litres of petrol; 2 litres of oil]

6. The input coefficient matrix in Nwafor-Ewee’s Bread Industry is given as:

[pic]

and the specific final demand vector as:

[pic].

Using Cramer’s Rule or otherwise, calculate the output matrix: x1, x2, and x3, required to satisfy the final demand vector.

[Answer: 150, 200, -350]

7. The charge for electricity is 3 ¢ per unit for lighting and ½ ¢ per unit for heating. A man’s bill for a quarter should have been $8. By mistake, his lighting was charged at ½ ¢ per unit and his heating at 3 ¢ per unit. The amount of the bill was $13. Use any matrix method to find how many units of electricity he used altogether.

[Answer: 600]

8. The input coefficient matrix in Malcolm’s Bread Industry is given as:

[pic]

and the specific final demand vector as: [pic].

Using Cramer’s Rule, find the output matrix x1, x2, x3, required to satisfy the final demand vector.

[Answer: – 1, – 6, 6]

9. FoodPerfect Corporation manufactures three models of the Perfect Foodprocessor. Each Model X processor requires 30 minutes of electrical assembly, 40 minutes of mechanical assembly, and 30 minutes of testing; each Model Y requires 20 minutes of electrical assembly, 50 minutes of mechanical assembly, and 30 minutes of testing; and each Model Z requires 30 minutes of electrical assembly, 30 minutes of mechanical assembly, and 20 minutes of testing. If 2500 minutes of electrical assembly, 3500 minutes of mechanical assembly, and 2400 minutes of testing are used in one day, use matrix method to find how many of each model will be produced.

[Answer: 40 Model X, 20 Model Y, 30 Model Z]

10. The TCI Housing Authority (TCI-HA) has decided to build 20 one-bedroom, 15 two-bedroom and 10 three-bedroom units of their new National Housing Scheme in Grand Turk. The building materials required for each type of house are cement, wood, glass, paint and labour given as:

Cement Wood Glass Paint Labour

One-bedroom 6 25 12 7 13

Two-bedroom 8 20 10 6 10

Three-bedroom 10 15 8 5 7

If cement costs $450 per unit, wood costs $500 per unit and glass, paint and labour cost $1000, $200 and $1200 respectively, calculate:

(a) The quantity of materials needed for the construction of the houses;

(b) How much TCI-HA needs to construct each house in the scheme;

(c) The total cost of building materials for the houses in the scheme.

[Answer: (a) 340 (cement), 950 (wood), 470 (glass), 280 (paint), 480 (labour);

(b) $44,200 (1-bedroom), $36,800 (2-bedroom), $29,400 (3-bedroom);

(c) $1,730,000]

11. A recreation center wants to purchase compact discs (CD’S) to be used in the center. There is no requirement as to the artist. The only requirement is that they purchase 40 rock CD’s, 32 western CD’s, and 14 blues CD’s. There are three different shipping packages offered by the company. They are an assorted carton, containing 2 rock CD’s, 4 western CD’s, and 1 blues CD; a mixed carton containing 4 rock and 2 western CD’s; and a single carton containing 2 blues CD’s. What combination of these packages is needed to fill the center’s order?

[Answer: 4 assorted cartons, 8 mixed cartons, 5 single cartons]

12. An art teacher finds that colored paper can be bought in three different packages. The first package has 20 sheets of white paper, 15 sheets of blue paper, and 1 sheet of red paper. The second package has 3 sheets of blue paper and 1 sheet of red paper. The last package has 40 sheets of white paper and 30 sheets of blue paper. If he needs 200 sheets of white paper, 180 sheets of blue paper, and 12 sheets of red paper, how many of each type of package should he order?

[Answer: 2 packages of 1st type, 10 packages of 2nd type, 4 packages of 3rd type]

13. The grocery store we use does not mark prices on its goods. My wife went to this store, bought three 1-pound packages of bacon and two cartons of eggs, and paid a total of $7.45. Not knowing she went to the store, I also went to the same store, purchased two 1-pound packages of bacon, and three cartons of eggs, and paid a total of $6.45. Now we want to return two 1-pound packages of bacon and two cartons of eggs. How much will be refunded?

[Answer: $5.56]

14. A citrus company completes the preparation of its products by cleaning, filling, and labeling bottles. Each case of orange juice requires 10 minutes in the cleaning machine, 4 minutes in the filling machine, and 2 minutes in the labeling machine. For each case of tomato juice, the times are 12 minutes of cleaning, 4 minutes for filling and 1 minute to label. Pineapple juice requires 9 minutes of cleaning, 6 minutes of filling, and 1 minute of labeling per case. If the company runs the cleaning machine for 398 minutes, the filling machine for 164 minutes, and the labeling machine for 58 minutes, how many cases of each type of juice are prepared?

[Answer: 20 cases of orange juice, 12 of tomatoes juice, 6 of pineapple juice]

15. A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for $75, main seats for $50, and balcony seats for $35. If all the seats are sold, the gross revenue to the theater is $23,000. If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is $21,500. How many are there of each kind of seat?

[Answer: 40 orchestra seats, 260 main seats, and 200 balcony seats]

16. A hospital dietician is planning a meal consisting of three foods whose ingredients are summarized as follows:

Units of

Food I Food II Food III

Units of Protein 10 5 15

Units of Carbohydrates 3 6 3

Units of Iron 4 4 6

Determine the number of units of each food needed to create a meal containing 100 units of protein, 50 units of carbohydrates, and 50 units of iron.

[Answer: 5 units of food I, 5 units of food II, 5/3 units of food III]

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