practiceExam1Sample3

practiceExam1Sample3 - Problem 1: [10 points] In each part...

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Problem 1 : [10 points] In each part of this problem, A , B , and are square matrices. C (a) Show that . 11 () AB B A −− = 1 (b) Use the results from part (a) to write down 1 ABC in terms of 1 A , 1 B , and 1 C . (c) True or false: If A λ is an eigenvalue of A , then 3 A is an eigenvalue of . State your reasoning. 3 A 1
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Problem 2 : [10 points] For each of the following ODEs, write down all of the terms from the list below that apply to the ODE: linear, nonlinear, first order, stiff (a) 3 dy yx dx =+ (b) 2 3 2 dy d y x y dx dx + (c) 3 dy x y dx (d) 3 dy x y dx = (e) 2 33 2 16 x x dy d y ex dx dx −− e 2
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Problem 3 : [10 points] In this problem you will discuss some properties of the following three matrices: 2 21 69 (1 ) , B= , C= 3 10.001 15 3 41 3 2 A ππ ⎛⎞ + ⎜⎟ = ⎝⎠ . (a) Calculate the determinant of each matrix. (b) What do the determinants of these matrices tell you about whether solutions can be found to the linear systems , 1 Ax b = 2 B xb = , and 3 Cx b = when [1, 1] T b = ?
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practiceExam1Sample3 - Problem 1: [10 points] In each part...

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