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q4 - Name Linear Algebra Quiz 4.2 1(20 points 4 −2 11(a...

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Unformatted text preview: Name: Linear Algebra Quiz 4.2 June 20, 2006 1.(20 points) 4 −2 11 (a) Compute the characteristic polynomial of A. (b) Compute the eigenvalues of A. (c) Find a basis for R2 consisting of eigenvalues of A. (d) Find an invertible matrix P and a diagonal matrix D such that P −1 AP = D. (e) Compute P −1 . (f) Write A as a product of P, P −1 , and D (in the correct order!). (g) Compute eA . (h) Find a basis for the space of solutions to the system of linear differential equations Given A = x1 (t) = 4x1 (t) − 2x2 (t) x2 (t) = x1 (t) + x2 (t) Solution: (a) Compute: λ2 − 5λ + 6 = (λ − 3)(λ − 2). (b) Read off λ = 2, λ = 3. 1 2 (c) Compute v1 = , v2 = . 1 1 12 20 (d) Set P = . Then P −1 AP = . 11 03 −1 2 (e) P −1 = . 1 −1 12 20 −1 2 (f) A = P DP −1 = . 11 03 1 −1 12 e2 0 −1 2 (g) eA = P eD P −1 = . 3 11 0e 1 −1 (h) Basis for space of solutions is given by the columns of etA = −e2t + 2e3t 2(e2t − e3t ) −e2t + 2e3t , in other words, { 2t 3t 2t 3t −e + e 2e − e −e2t + e3t the space of solutions. 1 12 e2t 0 −1 2 = 3t 1 −1 11 0e 2(e2t − e3t ) , } is a basis for 2e2t − e3t ...
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