05fp - Tuesday 31 May 2005 Name Math 5A Final Exam Review...

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Unformatted text preview: Tuesday, 31 May 2005 Name: Math 5A, Final Exam Review 105 1. Let A = 2 1 6 . 014 a. Compute det A. b. Compute A−1 . Verify that this is, indeed, the inverse of A. 5 6 . c. Solve Ax = b, where b = 7 2. Suppose for a mass-spring system, one obtains the differential equation x + 4x + 5x = 0, with initial values x(0) = 1 and x (0) = 0. a. Solve the equation. Write it in its phase-amplitude form. b. What is the earliest time at which this system crosses its equilibrium position? 3. a. Solve y − 4y = 6e2t . b. Write the aforementioned differential equation as an equivalent system of first order differential equations. 4. Given: dx dt dy dt = 5x + 4y, x(0) = 1 = −4x + 5y, y (0) = −3 a. Solve it. b. What is the behavior of its equilibrium point (0,0)? 5. Given: dx dt dy dt = 3 sin x + y 3 =y a. Find its points of equilibrium. b. Linearise the system at (0,0). Determine its stablility. 6. True or False? a. AB = BA for any matrices A, B. b. det AB = (det A)(det B) c. If A2 = 0, then A = 0. d. Let c be a scalar. If λ is an eigenvalue for A, then cλ is an eigenvalue for cA. ...
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