(Penalty 10 points for missing staple.) [First Name] Math 2333: [Last Name] THQ 12, 20-04-17, 19:14 [Net ID] Math 2333: THQ 12 Instructions: 1. Fill in the requested information on the line above. 2. This handout is due at the beginning of class on Monday 3. This handout must be printed out and stapled. You may print it double sided. 4. There will be a one point penalty for each minute for late submissions. 5. Your work must be hand written on this handout. 6. Refer to the textbook for definitions and problem statements. 7. You must show all work and check your answers. You may receive zero or reduced points for insufficient work. 8. Your work must be neatly organized and written. You may receive zero or reduced points for sloppy work. 9. Sometimes only a subset of these questions may be graded. You will not be told which questions will be graded in advance. 10. Students are not allowed to ask the Math Lab for assistance for any homework assignments. [Section] Page 2 of 8 Math 2333: Question 1 2 3 4 5 6 Total Points 10 10 10 10 10 10 60 THQ 12, 20-04-17, 19:14 Score 1. (10 points) Find the LU-decomposition of the following matrix, and then use that to find the determinant. 2 3 2 A= 8 9 5 4 6 7 (a) L = (b) U = (c) det A = Page 3 of 8 Math 2333: THQ 12, 20-04-17, 19:14 2. (10 points) Given the following LU-decomposition A = LU , answer the given questions. 6 2 3 A= 6 5 2 18 8 4 1 0 0 L= 2 1 0 6 4 1 3 6 2 U = 0 7 2 0 0 8 (a) What is the determinant of A? Answer: det(A) = 5 (b) What is the solution to Ax = b, where b = 2 . 4 Answer: x = Hint: Solve LU x = b from the outside in, solving Ly = b first, then solve U x = y. Show work on separate sheet (see the next page) and copy your final solution here. Page 4 of 8 Show work for previous question here. (c) Solve Ly = b: (d) Check: (e) Solve U x = y: (f) Check: (g) Check original system, Ax = b: (h) Copy your answer to previous page. Math 2333: THQ 12, 20-04-17, 19:14 Page 5 of 8 Math 2333: THQ 12, 20-04-17, 19:14 3. (10 points) Find the LU-decomposition of the following matrix, and then use that to find the determinant. Note: You should expect fractions to occur. 3 5 9 A= 4 6 1 1 3 5 (a) L = (b) U = (c) det A = Page 6 of 8 4. (10 points) Maximize f = 4x + 3y, subject to the constraints: 3x + 5y 6 30, 6x + 2y 6 24, x > 0, y > 0. You may use any valid method. Math 2333: THQ 12, 20-04-17, 19:14 Page 7 of 8 Math 2333: THQ 12, 20-04-17, 19:14 5. (10 points) Set up the Initial Simplex Tableau to maximize f = 3x + 5y + 4z subject to: 3x + 5y + 2z 6 92, 7x + 8y + 5z 6 76, 6x + 4y + 3z 6 81, x > 0, y > 0, z > 0. Note: You do not need to solve this, just set up the starting matrix. (a) Write the Initial Simplex Tableau: (b) Which is the first column that would contain a pivot element? (c) In which row would that pivot element be chosen? (d) Circle the first pivot element in the Initial Simplex Tableau you wrote above. (e) Again, do not complete the simplex algorithm: you are done! Page 8 of 8 Math 2333: THQ 12, 20-04-17, 19:14 6. (10 points) The following is the Final Simplex Tableau for an optimization problem. The main variables are x, y, and z, and the slack variables are u, v, and w (for two constraints). Analyze the Final Simplex Tableau and give values of x, y, and z to optimize the objective f . The columns represent x, y, z, u, v, w, f and the numerical value, respectively. 3 4 1 4 1 2 5 4 0 1 1 3 3 0 4 0 0 2 3 1 3 4 1 0 1 3 0 1 2 0 0 5 3 0 1 4 x= y= z= f (x, y, z) = 0 23 0 17 0 32 1 573