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131F06.probsess1 - Compute A 2 Without doing any further...

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Problem Session for MTH G131: 10/12/06 1). Find eigenvalues and eigenvectors of the matrix A = 3 - 5 2 - 3 Use your result to find the solution of the following ODE: d dt x y = A x y , x (0) y (0) = 1 2 2). For a 2 × 2 matrix A with eigenvalues λ 1 and λ 2 show that Trace( A ) = λ 1 + λ 2 , Det( A ) = λ 1 λ 2 3). Suppose the 2 × 2 matrix A has distinct real eigenvalues 0 < λ 1 < λ 2 , with eigenvectors v 1 and v 2 . Show that for most vectors w lim n →∞ λ - 1 2 A n w c v 2 for some constant c . For which vectors w is this result false? 4). Consider the 2 × 2 matrix A = 0 1 1 0 Compute
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Unformatted text preview: Compute A 2 . Without doing any further calculations, what can you con-clude about the eigenvalues of A ? 5). Consider the 3 × 3 matrix B = 1 1 1 Compute B 3 . Without doing any further calculations, what can you con-clude about the eigenvalues of B ? 6). Suppose the matrix A satisfies the equation A 4-3 A 3 + 6 A 2-I = 0 What can you conclude about the eigenvalues of A ?...
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