ma527PracticeFinal2011

ma527PracticeFinal2011 - MATH 527 PRACTICE PROBLEMS 1....

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MATH 527 PRACTICE PROBLEMS 1. Which of the following are vector spaces? i) The set of all 3x3 matrices A such that det A =0 . ii) The set of all 2x2 matrices A such that A ± 12 34 ² = ± ² A . iii) The set of all symmetric 3x3 matrices. A. iii) only B. i) and ii) C. i) and iii) D. ii) and iii) E. i), ii), and iii) 2. Which of the sets of vectors are linearly independent? i) (0 , 0 , 1), (0 , 1 , 1), (0 , 3 , 2) ii) (1 , 2 , 3), (4 , 5 , 6), (7 , 8 , 9) iii) (0 , 0 , 0), (0 , 1 , 0), (0 , 0 , 1) A. i) B. ii) C. iii) D. i) and iii) E. None 3. The inverse of the matrix ± 2 1 8 5 ² is A. ± 5 1 4 1 ² B. 5 2 1 42 ! C. 5 2 1 2 41 ! D. 5 2 1 2 4 1 ! E. Matrix has no inverse. 4. Suppose that the system Ax = b ,whe re A is an n × n matrix, has no solutions. Which of the following are true? i) The homogeneous equation Ax = 0 has inFnitely many solutions. ii) The rank of A is less than n . iii) A has no inverse. A. iii) only B. i) and ii) C. i) and iii) D. ii) and iii) E. i), ii), and iii) 1
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2 MA 527 FINAL EXAM 5. The rank of the matrix 01 20 02 40 0 3 60 is A. 0 B. 1 C. 2 D. 3 E. 4 6. The eigenvalues for the matrix 101 020 363 are A. 1, 2, 3 B. 1, 2, 0 C. 2, 3, 4 D. 1, 3, 0 E. 2, 4, 0 7. The eigenvalues of 0 10 11 1 0 are2 ,0 ,and 1. An eigenvector correspond- ing to 1is A. 1 0 1 B. 1 0 1 C. 1 1 1 D. 1 2 1 E. 0 0 0
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MA 527 FINAL EXAM 3 8. One solution to y 0 = ± 30 12 ² y is y = ± 0 e 2 t ² . Another linearly independent solution is A. ± e 2 t e 3 t ² B. ± 0 e 2 t + e 3 t ² C. ± e 3 t 0 ² D. ± e 3 t e 2 t ² E. ± e 3 t e 2 t + e 3 t ² 9. For the system y 0 1 = y 1 +3 y 2 y 0 2 =4 y 1 +2 y 2 the origin is A. an unstable node B. a stable node C. a saddle point D. a stable spiral point E. an unstable spiral point 10. For the system y 0 1 =6 y 1
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ma527PracticeFinal2011 - MATH 527 PRACTICE PROBLEMS 1....

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