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math34600fall2005

# math34600fall2005 - Math 34600 Final Exam Fall 2005...

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Math 34600 Final Exam Fall 2005 Instructions: Answer any seven questions. Omit one. 1a) Find the inverse of the matrix . 1 1 0 1 2 1 0 1 2 A = b) Using solve the system of equations: . 1 A 1 2 2 2 2 x y x y z y z = + = + = − 2a) Let . Find . 0 2 1 1 1 0 1 0 0 1 0 1 2 3 0 3 B = det( ) B b) Find det( . c) Find det(3 . 2 ) B ) B 3a) Find all solutions of the system . 0 4 2 5 2 x y z x y z x y z + + = + = + = 3 3 } 2 b) Explain why you cannot use Cramer's Rule to answer problem 3a). 4) Find all eigenvalues and all associated eigenvectors for the matrix . 1 2 2 1 A = 5) Let P ± . Which of the following subsets is a subspace of ? { 2 quadratic polynomials = 2 P a) {quadratic polynomials with }. ( ) p x (0) 1 p = b) {quadratic polynomials with }. ( ) p x (1) 0 p = c) For each of the subsets that are subspaces, find a basis for the subspace. 6) Prove or disprove the following statements about matrices A and B . n n × a) If A and B are invertible, so is . AB b) If A and B are invertible, so is . A B + c) If λ is an eigenvalue for A , then is an eigenvalue for . 2 λ 2 A 7) Let

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math34600fall2005 - Math 34600 Final Exam Fall 2005...

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