pset6 - 18.06 Problem Set 6 Due Wednesday 9 April 2008 at 4...

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18.06 Problem Set 6 Due Wednesday, 9 April 2008 at 4 pm in 2-106. Problem 1: a) Do problem 2 from section 6.1 (pg. 283) in the book. b) Do problem 9 from section 6.1 (pg. 284). Problem 2: Do problem 13 from section 6.1 (pg. 285) in the book. Problem 3: Consider the matrix M = 2 2 1 1 - 14 - 6 - 9 - 7 - 2 - 1 - 2 - 1 8 1 7 4 a) One eigenvector is x 1 = (1 , 1 , 0 , - 3). What is the corresponding eigenvalue? b) Note that det( M ) = 0. Use this information to find another eigenvalue λ 2 - how do you know this must be an eigenvalue? c) A third eigenvalue is λ 3 = - 1. Write down (but don’t solve) a linear system that can be solved to find x 3 . d) What is the fourth eigenvalue? (Hint: use the trace.) Problem 4: a) Do problem 8 in section 6.2 (pg. 299) b) Do problem 18 in section 6.2 (pg. 300) Problem 5: Here’s an example of an invertible 3 by 3 matrix with only 2 different eigenvalues: A = 4 1 - 1 2 5 - 2 1 1 2 a) Find the eigenvalues of A . b) Find 3 linearly independent eigenvectors of
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pset6 - 18.06 Problem Set 6 Due Wednesday 9 April 2008 at 4...

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