MATH 260 LINEAR ALGEBRA EXAM I Fall 2011

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1 MTH 60 LINER LGEBR EXM I Fall 011 Instructions: Do not write anything besides your name on this page of the exam. Write all work and answers in the space provided on pages -18. If you submit work for credit on additional sheets of paper, I will reduce your overall score by 5%. For all questions, I reserve the right to apply the no work, no credit policy at my discretion, so make sure you write your work carefully and clearly. There are twelve regular (mandatory) questions, each worth ten (10) points, of which you must do exactly ten. You must clearly mark the two questions you want to skip "DO NOT GRDE" or something equally clear, or the two extra questions may be counted against you. s well, you may submit up to two of the (optional) bonus questions, worth 10 (ten) points each. Your score will be calculated as a fraction of 100. In addition, you will not be permitted to leave the room until you are finished with this exam. If you leave prior to submitting your exam, you need not bother to return, as you will be done. Best of luck 1) a) [3 points] Joe is a meat & seafood supplier. He supplies beef, chicken, and pork to two customers: Seoul Buffet, and NY Mongolian BBQ. Seoul Buffet orders 60 lbs of beef, 70 lbs of chicken, and 50 lbs of pork. NY Mongolian BBQ orders 50 lbs of beef, and 50 lbs of chicken. Store this information in a 3 matrix, and clearly label the rows and columns. Call this matrix. b) [4 points] Joe can fill his orders from four different wholesalers: Cargill (C), Hormel (H), Smithfield (S), and Tyson (T). The cost of the products they sell ($/lb) is given in the matrix B below. What is the practical meaning of the product B? Find the product using your calculator and explain. C H S T Beef Chicken Pork c) [3 points] Suppose Joe doesn t care which wholesaler he buys from, so long as his cost is minimized. Which wholesaler will he use? Justify your answer briefly. ) Use Gauss-Jordan Elimination to solve the following system: x 3x x 3x x 4x 3 5x 8x 7 1 Math 60 Exam 1 Fall 011/Page 1 of 5.

2 3) Use Gaussian Elimination to solve the following system: x 3x 3x 17 x x 3x 1 3x x x 4) Given the matrices and B below, express B as a linear combination of columns of , B ) Suppose, B, C, and D are matrices with the following sizes: is 3; B is 3 ; C is 3 3; and D is. For each part, determine whether the matrix expression is defined. If it is, give the size of the resulting matrix. a) B D b) B C c) e) B T C d) DB T T BC 6) Let a) Compute 3 b) Compute ) Let 0 0 1, B 0 0 1, and C Find elementary matrices E 1, E and E 3 such that: a) E 1 B b) EE 3 C Math 60 Exam 1 Fall 011/Page of 5.

3 8) Solve the system by expressing it in the form x b and making use of 4x13x x3 16 x15x 3x3 8 4x x x ) If a) c) e) find: b) d) n 1 n 10) Determine whether the following statements are true or false. If the statement is true, justify why it is true. If the statement is false, explain why, or provide a counter-example. a) Say is an n n matrix which is not invertible. Then x 0 has infinitely many solutions. b) Say is an n n matrix which is symmetric and invertible. Then also symmetric. 3 11) Find the cubic polynomial p x ax bx cx d that contains the points 1,1, 0,1, 1,1, and 1 is,7 by setting up and solving the appropriate linear system. You may use your calculator to solve the system once it is set up. Math 60 Exam 1 Fall 011/Page 3 of 5.

4 1) Model the flow of current in the electrical system below using a system of linear equations. Write the augmented matrix of the system. You may use your calculator to find the reduced row echelon form for this augmented matrix. Interpret this matrix to determine the currents for the given electrical network. If necessary, write the currents as reduced fractions. I1 4 volts I1 ohms I ohms I B 5 ohms I3 I3 8 volts 13) BONUS [10 points] Write as a product of elementary matrices ) BONUS [10 points] n out-of-shape athlete can run at 6 mph, swim at 1 mph, and bike at 10 mph. He enters a triathlon, which requires all three events, and finished successfully after 5 hours and 40 minutes. friend of his, in somewhat better condition, can run at 8 mph, swim at mph, and bike at 15 mph, was also able to finish successfully in 3 hours and 35 minutes. If the total course was 3 miles long, how many miles was each segment (running, swimming, and biking)? Set up and solve the appropriate linear system. You may use your calculator to solve the system once it is set up. 15) BONUS [10 Points] Determine whether the following statements are true or false. If the statement is true, justify why it is true. If the statement is false, clearly explain why, or better still, provide a counter-example. a) Say and B are both n n B are symmetric. b) Say and B are both n n and B are upper triangular., such that, such that B is symmetric, then and B is upper triangular, then Math 60 Exam 1 Fall 011/Page 4 of 5.

5 16) BONUS [10 Points] Determine whether the following statements are true or false. If the statement is true, justify why it is true. If the statement is false, clearly explain why, or better still, provide a counter-example. a) Say is m n, with m n. Then x b is inconsistent. b) Say is m n, with m n. Then x b is dependent. Math 60 Exam 1 Fall 011/Page 5 of 5.