APPLIED LINEAR ALGEBRA ASSIGNMENT

QUESTION 1 -

1. (a) How would you write the matrixin Matlab/Octave?

(b) Give the Matlab/Octave command to find the transpose of A.

(c) Give the command to find the entry of A in the second row and second column.

2. Set up any 3 x 3 matrix and call it A Then write some Matlab/Octave command-line statements to perform the following operations.

(a) interchange cohinnis 2 and 3.

(b) add a fourth column (of 0s)

(c) remove the second column

3. We can draw the Mexican hat using the following equation

z = sin(r)/r, where r = √(x²+y²) + ε, and ε = 2.2204 x 10^-16

Write the Matlab/Octave code that solves for z and plot z in the interval -8 < x < 8, -8 < y < 8

Hint: (Plot using the meshgnd function with the grid width of 0.5)

4. Write a Matlab/Octave code that draws a graph of the population of the United States from 1790 to 2000, using the (logistic) model

p(t) = 197273000/1 + e^{-0.03134(t-1913.25)}

where t is the date in years. The actual data (in 1000s) for every decade from 1790 to 1950 is as follows:

Superimpose the data on the graph of P(t).

QUESTION 2 -

1. On Matlab/Octave command line, assign values to the variables a and b, for example a = 3, and b = 5, and write statements to find the sum, difference, product, and quotient of a and b.

2. What do the following Matlab/Octave statements do?

(a) 1 + 1  5

(b) 1+ [1 · 5]

(c) 1  5'

(d) [1  5]'

3. Given the following vectors

a = [2   4   5],

b = [6   2   2],

Calcuate-

 (a) a. *b,

(b) a. /b,

(c) a. ^b,

4. Given the vector [1 2 3 4 5] Write Matlab/Octave statements that use different vector array operations to produce the following vectors:

(a) [2  4  6  8  10]

(b) [½  1  3/2   2  5/2]

(c) [1   ½  1/3  ¼   1/5]

(d) [1  1/2²  1/3²  1/4²   1/5²]  

QUESTION 3 -

1. What does the following Matlab/Octave command do?

(a) plot(x, y, '- -')

(b) plot(x, y, '0')

(c) plot(x, sin(x), x, cos(x), 'om- -')

2. Write a Matlab/Octave code to evaluate the following series _n=1∑^∞U_n in which U_n is not known explicitly but is given in terms of a recurrence relation. You should stop the summation when |U_n| < 10^-8. U_n+1 = U_n², with U₁ = 0.5.

3. (a) How do you find the dominant eigenvalues and eigenvectors of the following matrix, using both eig and power method. Also give the power method code

(b) Also give the power method code

(c) How can one diagonalize matrix A?

QUESTION 4 -

1. The Milko Dairy can receive no more than 100,000 litres of milk per day. Due to a long-term contract, at least 10,000 litres of each day's milk must be used for chase manufacture. The balance can be used for bottled milk butter or yoghurt. At today's market prices, the contribution to profit and fixed cost of each litre of milk, when put to these uses, is as follows

Butter                   R2 50

Cheese                 R2 00

Bottled Milk           R1 50

Yoghurt                R3.00

The butter equipment can handle up to 50,000 litres of milk per day, and the milk equipment up to 40,000 litres. Part of the yoghurt and cheese processing uses the same equipment and this imposes a limit on the combined usage of 50,000 litres per day. The butter and cheese packagng equipment can handle a cornbmed usage of at most 55,000 litres per year.

What mix of products should the company produce so as to maximize profit?

2. Find the maximum value as well as the point at which the maximum occurs of

L = 2x₁ + 3x₂ + 4X₃ + 3x₄

subject to the constraints.          

15x₁ + 2x₂ + 15X₃ + x₄ ≤ 30

1x₁ + 2x₂ + 1x₃ + 3x₄ ≤ 45

5x₁ + 4x₂ + 7x₃ + 2x₄ ≤ 65

6x₁ + 3x₂ + 7x₃ + 4x₄ ≤ 60

8x₁ + 4x₂ + 8x₃ + 2x₄ ≤ 70

x₁, x₂, x₃, x₄ ≥ 0