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Ordinary Differential Equations MATH 308 - 523 A. Bonito April 14 Spring 2015 Last Name: First Name: Quiz 5 • 5 minute individual quiz; • Answer the questions in the space provided. If you run out of space, continue onto the back of the page. Additional space is provided at the end; • Show and explain all work; • Underline the answer of each steps; • The use of books, personal notes, calculator, cellphone, laptop, and communication with others is forbidden; • By taking this quiz, you agree to follow the university’s code of academic integrity. Exercise 1 100% Find the eigenvalues and one associated eigenvector (for each eigenvalue) of the matrix below and determine whether they are linearly independent. −3 3/4 −5 1 Ordinary Differential Equations MATH 308 - 523 A. Bonito April 14 Spring 2015 Quiz 5: solutions Exercise 1 100% The eigenvalues are given by det(A − λI) = 0. This is det −3 − λ −5 3/4 1−λ =0 or computing the determinant λ2 + 2λ + 3/4 = 0 The solutions of the above equation are given by λ1 = − 3 λ2 = − . 2 1 2 The eigenvectors ξ 1 = (ξ1 , ξ2 )t associated with λ1 satisfy 3 −5ξ1 + ξ2 = 0 2 or ξ1 = α 1 3 − 10 10 −3 for any constant α. For instance ξ1 = The eigenvectors ξ 2 = (ξ1 , ξ2 )t associated with λ2 satisfy 3 −3ξ1 + ξ2 = 0 2 or 2 ξ =α 1 1 2 for any constant α. For instance, 2 ξ = 2 1 . The two eigenvectors are independent because they correspond to different eigenvalues.