math 235 ch 7 practice

math 235 ch 7 practice - Math 235 Section 6 Chapter 7...

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Unformatted text preview: Math 235 - Section 6, Chapter 7 Practice Problems Adam Gamzon 1. Let A= (a) Find the eigenvalues of A. (b) Find eigenvectors v1 , v2 corresponding to the eigenvalues found in part (a). (c) What is the matrix of A with respect to the basis B = {v1 , v2 } found in part (b)? 2. In an unfortunate accident involving an Austrian truck, 100 kg of a highly toxic substance are spilled into lake Sils in the Swiss Engadine Vally. The river Inn carries the pollutant down to Lake Silvaplana and later to Lake St. Moritz. After t weeks, let x1 (t) be the pollutant in Lake Sils, x2 (t) be the pollutant in Lake Silvaplana and let x3 (t) be the pollutant in Lake St. Moritz. Set x1 (t) x(t) = x2 (t) . x3 (t) Suppose that we estimate that the pollutant in the lakes to change according to the model 0.7 0 0 x(t + 1) = 0.1 0.6 0 x(t). 0 0.2 0.8 (a) Find eigenvalues and corresponding eigenvectors of the coefficient matrix of the system. (b) Use part (a) to find formulas for x1 (t), x2 (t) and x3 (t). Hint: we know that x1 (0) = 100, x2 (0) = 0 and x3 (0) = 0. (c) When does the pollution in Lake Silvaplana reach a maximum? (d) What happens to x1 (t), x2 (t) and x3 (t) as t → ∞. 3. Find a formula for At where A = 12 . 36 24 . 33 1 ...
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This note was uploaded on 11/22/2011 for the course MATH 255 taught by Professor Staff during the Spring '10 term at UMass (Amherst).

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