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Math 54 - Quiz 09 Ans, Spring 2006

Math 54 - Quiz 09 Ans, Spring 2006 - Therefore the general...

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Math 54, Quiz 9 Solutions Sections 202/203 GSI Carter 1. Write the word “true” or “false”: Let A be a 3 × 3 matrix with real entries and complex eigenvalue α + βi (that is, α is a real number and β is a nonzero real number). Then A has three distinct eigenvalues. True. The roots of the characteristic polynomial of A must be α + βi , α - βi , and some real number. 2. (a) Solve the following initial value problem: x = 2 4 4 2 x x (0) = 0 1 The characteristic polynomial of the matrix is ( λ - 2) 2 - 16 = λ 2 - 4 λ - 12 = ( λ + 2)( λ - 6) , so the eigenvalues are - 2 and 6. The eigenspace for - 2 is the null space of - 4 - 4 - 4 - 4 which is spanned by (1 , - 1) T . The eigenspace for 6 is the null space of 4 - 4
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Unformatted text preview: . Therefore the general solution to the system of diﬀerential equations is given by y = c 1 ± 1-1 ² e-2 t + c 2 ± 1 1 ² e 6 t . From the initial condition, we have that ± 1 ² = c 1 ± 1-1 ² + c 2 ± 1 1 ² , and therefore c 1 =-1 2 and c 2 = 1 2 . Therefore the solution to the initial value problem is given by y =-1 2 ± 1-1 ² e-2 t + 1 2 ± 1 1 ² e 6 t . (b) What type of system is the above diﬀerential equation? Circle one. i. source ii. sink iii. saddle point iv. spiral in v. spiral out vi. periodic Since the eigenvalues are real and have opposite signs, it is a saddle point. 1...
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