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MATH 307 FALL 2002 EXAM 2 1. (30 points) Let −3 1 4 A = −3 1 3 −1 1 2 The characteristic polynomial of A is fA (λ) = λ3 −3λ+2. Find all eigenvalues and eigenvectors of A, including the algebraic and geometric multiplicity in each case. Is A diagonalizable? 2. (25 points) Find a basis for the kernel and image of 1 1 2 3 3 A= 2 2 1 0 −3 12 and state the dimension of each. Is ~b = in the image? −17 3. (15 points) Let 1 A= k 0 2 0 3 0 0 4 For what values of k are the eigenvalues of A all real? 4. (30 points) True or false? Explain your answer. a) If A, B are n × n matrices and A is singular then AB is singular. b) If A is invertible then it is diagonalizable. c) If ~v is an eigenvector of A then it is also an eigenvector of A 2 + I. d) If vectors ~v1 , ~v2 , ~v3 span a subspace V of Rn then V has dimension 3. 0 1 1 0 have the same eigenvalues. and e) The two matrices 0 1 1 0 f) The following vectors form a basis of R3 : 2 5 1 0 7 4 0 0 4