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Final10 - MA366 Make-up Final Last Name First Name Show all...

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MA366 Make-up Final Last Name: First Name: Show all work. A correct answer without supporting work is worth NO credit! (Some calculators can solve differential equations.) There should be no “hard” integrals, unless you mess up somewhere. If this happens, just leave it as an integral and explain how to finish the problem. 1
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2 (1) The following vectors X 1 and Y 1 are eigenvectors for a certain 3 × 3 matrix A corresponding to the eigenvalues 3 and 1 + 2 i respectively. Find the general solution to the system X 0 = AX in real form . No complex numbers allowed! 5 pts. X 1 = 0 3 1 , Y 1 = i - 1 i - 2 .
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3 (2) Given that X 1 , X 2 and X 3 are eigenvectors for the following matrix, find the general solution to X 0 = AX . Hint: To find the eigenvalue, compute AX i . 5 pts. A = - 3 6 - 3 4 - 3 2 6 - 12 6 X 1 = 1 0 - 2 X 2 = - 1 1 2 X 3 = - 1 2 5
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4 (3) The characteristic polynomial for the following matrix is p ( r ) = - ( r - 3) 2 ( r - 1) and the vector Y 1 is an eigenvector corre- sponding to
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