Solve. the following questions as attached below.

b) Given a matrix Aman with m # n, write an algorithm / pseudo code to determine the singular value decomposition, using only the function [ E, V] = cigs( A) which gives the eigenvectors of A in E and eigenvalues in the diagonal matrix V. (2) Pages 12-15 Q4 a) Solve by Cholesky's method, the following system. (3) 4x1 + 2ry + 1423 = 14 2x1 + 1717 - 5x3 = -101 14x1 - 5ry + 83xa = 155 b) Write a suitable numerical scheme for which the Gauss Seidel method for the following system converges. What are the relevant norms (just write in terms of the matrices and do not compute them) and their bounds that would ensure convergence? Perform two iterations using the starting value as (1, 1, 1). (2) 4x1 + 513 = 12.5 Il + 62 + 2rs = 18.5 8x1 + 2 + 1es = -11.5 Pages 16-17 Q5 a) How many additions, multiplications and divisions are involved in computing the largest eigenvalue (in absolute value) using the power method with scaling? Assume that the given matrix A is of size a x n and that we do m steps. , may be taken as the initial vector satisfying |zoll S dgm+. (2) b) Under what conditions on the matrix A, would we be able to compute the error o in approximationg the eigenvalue that is largest in absolute value using the power method? What happens when the initial starting vector x is an eigenvector of AT (2) c) For a vector y e R", prove that llyla S vmlly|s- (1) Pages 18-20 Q6a) The Town of Ghosts has the following requirement of police officers during the various time periods given in the table below: Period No. Time period Minimum Requirements 6 am - 10 am 22 10 am - 2 pm 55 2 pm - 6 pm 88 6 pm - 10 pm 110 10 pm - 2 am 44 2 am - 6 am 33 The duty for each of the Police Officers commence at 6 am or 10 am or 2 pm, .. . or I am and they work for 8 consecutive hours. Every Police Officer joins duty during any of these timings. Formulate an LPP for the minimum number of total police officers required for the town. (3) b) Assume that there are there are just three currencies, US Dollars, British Pounds and Euro and initially there is $10000 to start with. If the conversion rate from currency i to currency / is denoted by ay, formulate this as a linear programming problem for the return in doing the currency arbitrage. (2) - All the best + 2/2Exercise 6: Vertical Differentiation Suppose a market featuring two goods that differ in quality. Good 1 is of quality s, and good 2 is of quality $2, with $2 > $1. There is a mass of n = 1 consumers, of different types, identified by a taste parameter 0 ~ Ujaj. Consumer's utility function is given by u(0) = 0s' - p if it buys one unit of good with quality s 0 otherwise In what follows, assume s = 1 and s2 = 2. In other words, firms cannot choose their quality, this is given and does not change. Firms cost functions are given by: C(q;) = csiq; for i = 1,2 where q, is the output of firm i. We will also conjecture that parameter values are such that there is full coverage. a) (4 points) Find the taste parameter o that identifies a consumer who is indifferent between buying a unit of good 1 or a unit of good 2. b) (4 points) Derive an expression for the demand functions of each good (i.e., the demand functions that each firm is facing separately). c) (4 points) Set up each firm's problem. Find the firms' reaction functions and solve for the equilibrium prices d) (4 points) Assume now that c = 0.1, a = 1 and b = 3. Find the prices, quantities and profits for each firm. What are the market shares for each firm? Verify that the market is indeed fully covered (Hint: Think on the condition that needs to be met so that the market is fully covered)Question 1: (8 Marks) Simplify the following linear system of equations and then obtain its solution; 2x1 + 5x2 + 7%g = 25, -5x1 + 7%2 + 2x3 = -4,*1 + 22x2 + 23x3 = 71 Question 2: (6 Marks) Find the inverse of A = 2 4 3 , using its Row-Reduced Echelon Form -12 N Question 3: (6 Marks) For a matrix A = * -, there exists a non-singular matrix P such that AP = PD where, D- 5 2.Obtain any P and further verify the result AP = PD. Question 4: (3+4+2+1=10 Marks) For Gax ( $1 93)_1 92 -912 -921 911 with A-(919n-912921) #0. Consider a linear system of equations Ax = b where Aman, Xax1 and bax1. Based on the normal equation A Ax = A"b= w, let xis be the least square solution such that Axis=bus with the error as cus (b-bus). (a) For the case when rank(AHA) =n, obtain the closed form expression for Xis with Aman; 23 For the case with A = 1 27 1b= 30 ; using the formula for Gas given above, obtain the following 34 vectors. (b.1) XLs (b.2) bus (b.3) eLsFurthermore, "The Question" is expected to reduce sales of other Reebok shoes, and the life of the shoe is expected to be only 6 years. Reebok's CFO, Kenneth Watchmaker, had compiled the following information surrounding "The Question" project: 1. The life of the project is 6 years. 2. The retail price of the shoe is estimated to be in the $130-$150 range; Reebok's wholesale selling price will be $100.00 3. The athletic shoe market is projected to reach $18 billion during Year 1 and is growing at a rate of 3% per year. The market share projections for Reebok's "The Question" are: Year 1, 1.70%; Year 2, 1.75%; Year 3, 1.60%; Year 4, 1.45%; Year 5, 1.35%; and Year 6, 1.25%. 4. In order to produce the shoe, Reebok will need to build a factory in New Delhi, India. This will require an immediate outlay of $150 million, which will be depreciated on a 39 year MACRS basis. The depreciation percentages for the first six years, respectively, are: 2.6%, 5%, 4.7%, 4.5%, 4.3%, and 4.0%. Reebok analysts estimate that the building will be sold for $102.35 million at the project's termination. Note that this "salvage value" is not taken into consideration when computing the annual depreciation charges here. 5. Reebok must also immediately purchase equipment costing $15 million. Freight and installation of the equipment will cost $5 million. The equipment and freight/installation costs will be depreciated on a 5-year MACRS basis. The depreciation percentages for the six years, respectively, are: 20%, 32%, 19%, 12%, 11%, and 6%. It is believed that the equipment can be sold for $3 million at the project's termination. 6. In order to manufacture "The Question," two of Reebok's working capital accounts are expected to increase immediately. The inventory balance is expected to increase by $60 million and the accounts payable account is expected to increase by $15 million. These balances will be maintained until the final year of the project, at which time they will be recovered. 7. Sales of "The Question" are expected to reduce sales of other Reebok basketball sneakers. Specifically, Reebok's other sneaker sales are expected to decrease by $170 million during each year of the project. Assume these lost sales have the same margins as The Question. 8. Variable costs of producing the shoe are expected to be 31% of the shoe's sales. Including salvage value in depreciation computations is done for financial reporting purposes. Here, we are concerned with the cash flow impacts of taxes; for fax purposes, it is not necessary for firms to incorporate salvage valuc. 2