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40 pts. Problem 1. Let A be the matrix " −2 −2 6 5 # A= . Find the characteristic polynomial of A and the eigenvalues of A by hand computation. 70 pts. Problem 2. In each part, you are given a matrix A and the eigenvalues of A. Find a basis for each of the eigenspaces. Determine if A is diagonalizable and, if so, find an invertible matrix P and a diagonal matrix D so that P −1 AP = D. [The use of a calculator is highly recommended. Say what you put into the calculator and what you got out.] A. The eigenvalues are 1 and 2 and the matrix is 2 −1 1 A= 1 0 2 . 1 −1 3 B. The eigenvalues are 1 and 2 and the matrix is 2 −1 1 1 1 A= . 0 0 0 2 70 pts. Problem 3. Consider the system of differential equations (3.1) x01 (t) = x1 (t) + x2 (t) x02 (t) = −2x1 (t) + 4x2 (t) This system can be written in matrix form as x0 = Ax where A is the matrix " # 1 1 A= . −2 4 This matrix is diagonalizable, in fact P −1 AP = D where " # " # 1 1 2 0 P = , D= . 1 2 0 3 [The use of a calculator is recommended on this problem. Say what you put into the calculator and what you got out.] 1 A. Use the information above to find the matrix etA . B. Find the solution of the system (3.1) for arbitrary initial conditions ( x1 (0) = x01 x2 (0) = x02 C. Find the solution of the system (3.1) for the initial conditions ( x1 (0) = 2 x2 (0) = −1 2 EXAM Exam 2 Math 3351, Fall 2010 Nov. 3, 2010 • Write all of your answers on separate sheets of paper. You can keep the exam questions when you leave. You may leave when finished. • You must show enough work to justify your answers. Unless otherwise instructed, give exact answers, not √ approximations (e.g., 2, not 1.414). • This exam has 3 problems. There are 180 points total. Good luck!