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# %%practicefinal - MATH 527 PRACTICE PROBLEMS 1 Which of the...

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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 1 2 3 4 = 1 2 3 4 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) 3. The inverse of the matrix 2 - 1 8 - 5 is 5 2 1 4 2 ! 5 2 - 1 2 4 1 ! 5 2 - 1 2 4 - 1 ! 4. Suppose that the system Ax = b , where A is an n × n matrix, has no solutions. Which of the following are true? i) The homogeneous equation Ax = 0 has infinitely many solutions. ii) The rank of A is less than n . iii) A has no inverse. 1

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2 MA 527 FINAL EXAM 5. The rank of the matrix 0 1 2 0 0 2 4 0 0 - 3 - 6 0 is A. 0 B. 1 C. 2 D. 3 E. 4 6. The eigenvalues for the matrix 1 0 1 0 2 0 3 6 3 are 7. The eigenvalues of 0 - 1 0 - 1 1 - 1 0 - 1 0 are 2, 0, and - 1. An eigenvector correspond- ing to - 1 is 1 0 1 - 1 0 1 1 1 1 1 - 2 1 0 0 0
MA 527 FINAL EXAM 3

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