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2003test2 - M346 Second Midterm Exam 1 Find all the...

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M346 Second Midterm Exam, October 23, 2003 1. Find all the eigenvalues of the following matrices. You do NOT need to find the corresponding eigenvectors. [Note: the answers are fairly simple, and can be obtained without a lot of calculation, using the various “tricks of the trade”.] a) 3 1 5 17 1 3 4 - 10 0 0 2 - 1 0 0 1 2 b) 1 2 3 4 0 2 1 2 3 . 2. The eigenvalues and eigenvectors of the matrix A = 0 - 1 - 1 - 1 0 - 1 - 1 - 1 0 are λ 1 = - 2, λ 2 = 1 and λ 3 = 1, and corresponding eigenvectors b 1 = 1 1 1 , b 2 = 1 - 1 0 and b 3 = 1 0 - 1 . (That is, the eigenvalue 1 has multiplicity two, and a basis for the eigenspace E 1 is { b 2 , b 3 } .) a) Solve the difference equation x ( n + 1) = A x ( n ) with initial condition x (0) = 3 0 0 (which equals b 1 + b 2 + b 3 , by the way). That is, find x ( n ) for every n . b) With the situation of part (a), identify the stable, unstable, and neutrally stable modes. What are the limiting ratios x
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