II.4 (15 points) 1 1 0 41 Let A= 1 3 -2 . [1 0 1 (a) Find the characteristic polynomial (its factored form will be sufficient), and find all eigenvalues of A.

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1. Suppose that 4 and -1 are the only eigenvalues of a 2 x 2 matrix A. Which one of the following statements is true? A) 41 - A is invertible, and there is a nontrivial solution to Av = -v. B) 41 - A is not invertible, and there is a nontrivial solution to Av = v. C) det(41+ A) = det(-I + A) = 0. (D) det(41 - A) = 0 and rank(A + 1) = 1. E) None of the above.

Version 1 MATH 133 Final Exam - April 18, 2008 II.2 (15 points) Consider the matrix M = [1 1 | 1 2 1 2 3 5 | 1 3 4 5 5 14 6 -3 71 -5 8 -6 11 -13 23 -7 9 . (a) Find the reduced echelon form of M, showing your work. (You should find that M has rank 3.)

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II.4 (15 points) 1 1 0 41 Let A= 1 3 -2 . [1 0 1 (a) Find the characteristic polynomial (its factored form will be sufficient), and find all eigenvalues of A.

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