APPM 2360 Midterm: appm2360summer2015exam3_sol
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On the front of your bluebook write: (1) your name, (2) your student id number, (3) your instructor"s name, and (4) a grading table. Text books, class notes, and calculators are not permitted. 0 8 (a) find all eigenvalues of the matrix. Hint: use your answer from (a). (c) find all eigenvectors of a associated with the eigenvalue of largest algebraic multiplicity. (d) give, with proof, the dimension of the eigenspace for the eigenvalue that you chose. Solution 1: (a) expanding along the second column, det(a i) = ( + 2)((7 )( 8 ) + 54) = ( + 2)( 2 + 2) We look at the solutions to (a + 2i)x = 0. 0 6 hence all eigenvectors for = 2 are in the form. 0 (d) we found that two vectors spans this eigenspace so the dimension of the eigenspace is 2 .
