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Unformatted text preview: Math 330 Exam 4 Outline Exam 4 for Math 330 will take place on Friday April 17. Calculators will not be permitted. Eigenvalues  15 points You be given a 3 3 matrix which has 3 distinct eigenvalues. The matrix will have sufficient zero entries so that the characteristic polynomial can be easily determined and factored. Here is an example: 1 A = 1 5 9 0 3 0 9 8 1 SOLUTION : det( A I ) = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1 5 9 3 9 8 1 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = (3 ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1 9 9 1 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = (3 ) bracketleftBig (1 ) 2 81 bracketrightBig = (3 ) bracketleftBig 80 2 + 2 bracketrightBig = (3 )(  10)( + 8) So the eigenvalues are 3,10, and 8. Diagonalizable or Not  15 points You will be given an n n matrix with n equal to 3,4, or 5. You will be told all the eigenvalues although the list of these will have length less than n . You job will be to decide if the matrix is diagonalizable. By looking at the dimensions of the various eigenspaces (and seeing if these add up to n or something less than n ) you can decide if the matrix is diagonalizable. Here are two examples using 3 3 matrices: 2 1. The following matrix has eigenvalues 1 and 3. Is it diagonalizable?1....
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 Spring '08
 Johnson,J
 Math, Linear Algebra, Algebra

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