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OLLSCOIL NA hEIREANN, GAILLIMH NATIONAL UNIVERSITY OF IRELAND, GALWAY SUMMER EXAMINATIONS, 2009 FIRST employ science & INFORMATION TECHNOLOGY EXAMINATION MATHEMATICS [MA150] MA153 | ALGEBRA prof D. Armitage Professor T. Hurley Dr. G. Ellis Dr. G. Pfeier Time allowed: iii hours. Answer six-spot questions. 1. (a) A small dairy uprises triplet cheeses: mild, standard and mature. The adjacent table coremarizes the amount of energy, milk and drive used to produce one box of each of the terce cheeses to shrinkher with the amount of these resources visible(prenominal) per day. Resource Mild Standard Mature day-to-day accessible Energy 2 kWh 3 kWh 2 kWh 100 kWh quest out 4 L 4 L 3 L 150 L Labour 3 h 4 h 6 h 170 h Solve a schema of linear equations to determine the achievement gures that would ensure all resources be used. (b) Give one encourage of t for which the system x + 2y = ty 2x + y = tx has innitely many another(prenominal) solutions. (Hint: It might help to rearrange the equa- tions.) p.t.o. 2. Let A := 0 B@ 2 -6 0 -2 -8 2 2 -8 0 1 CA and B := 0 B@ 4 0 -3 1 0 -1 8 1 -7 1 CA : (a) elaborate AB and A-1 and then solve 2x - 6y = -4 2x + 8y - 2z = 2 2x - 8y = -10 : (b) Using (AT )-1 = (A-1)T or otherwise, solve 2x - 2y + 2z = 2 -6x - 8y - 8z = 4 2y = 2 : (c) stupefy a ground substance E much(prenominal) that EA is the result of replacing the rst wrangling of A by the sum of the rst and third rows of A. 3. (a) Calculate the determinant of the hyaloplasm A := 0 B@ 1 2 3 2 3 + t 10 2 4 t + 4 1 CA and determine all value of t for which A is NOT invertible. (b) Let f(t) be the vault of promised land of the parallelogram with vertices v = (1; 2t),w = (2t; 1), (0; 0) and v + w = (2t + 1; 2t + 1). Find a formula for f(t). (c) Find a 22 matrix A such that f : R2 ! R2; v 7! Av represents a clockwise rotation through 45o about the origin. 4. (a) Find three vectors p; v ;w 2 R4 such that the solutions to the syste! m w + 3x + 3y + 2z = 1 2w + 6x + 9y + 5z = 5 -w - 3x + 3y = 5 come the form p + v + w...If you want to get a full essay, order it on our website: BestEssayCheap.com

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