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mat1302W10-final(2) - University of Ottawa Department of...

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University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Professor: Aziz Khanchi Final Exam – Solutions April 2010 Surname First Name Student # Seat # Instructions: (a) You have 3 hours to complete this exam. (b) This exam consists of two parts. Questions 1 through 14 are answer only . For these questions, only your final answer will be considered for marks. Questions 15 through 22 are long answer . For these questions, you must show your work and justify your answers to receive full marks. Partial marks may be awarded for making sufficient progress towards a solution. (c) Each answer only question is worth two points. For the long answer questions, the number of points available for each question is indicated in square brackets. (d) All work to be considered for grading should be written in the space provided. The reverse side of pages is for scrap work. If you find that you need extra space in order to answer a particular question, you should continue on the reverse side of the page and indicate this clearly . Otherwise, the work written on the reverse side of pages will not be considered for marks. (e) Write your student number at the top of each page in the space provided. (f) No notes, books, scrap paper, calculators or other electronic devices are allowed. (g) You may use the last page of the exam as scrap paper. Good luck! Please do not write in the table below. Question 1–14 15 16 17 18 19 20 21 22 Total Maximum 28 4 4 4 6 8 6 6 4 70 Grade
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MAT 1302A Final Exam – Solutions Part A: Answer Only Questions For Questions 1–14, only your final answer will be considered for marks. Each question is worth 2 points. Question 1. What is the determinant of the matrix 4 7 10 - 4 0 0 1 / 2 - 6 7 8 0 0 3 4 - 5 / 2 0 0 0 - 2 7 / 3 0 0 0 0 1 ? Answer: - 12 Question 2. Give the characteristic polynomial of the matrix 2 0 0 0 - 5 1 0 0 7 0 2 0 5 - 6 3 0 . Answer: λ ( λ - 1)( λ - 2) 2 Question 3. Suppose A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 , and det A = 3. If B = a 21 a 22 a 23 a 11 a 12 a 13 a 31 a 32 a 33 , and C = a 11 a 12 a 13 a 21 a 22 a 23 a 31 - 3 a 11 a 32 - 3 a 12 a 33 - 3 a 13 , what are det B and det C ? Answer: det B = - 3, det C = 3 Question 4. Is 1 + i 1 an eigenvector of the matrix - 1 2 - 1 1 ? If so, what is the corre- sponding eigenvalue? Answer: Since - 1 2 - 1 1 1 + i 1 = 1 - i - i = - i 1 + i 1 , the answer is YES and the eigenvalue is - i .
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