# E4out - Math 330 Exam 4 Outline Exam 4 for Math 330 will...

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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- λI ) = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1- λ 5 9 3- λ 9 8 1- λ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = (3- λ ) vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle 1- λ 9 9 1- λ vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = (3- λ ) bracketleftBig (1- λ ) 2- 81 bracketrightBig = (3- λ ) bracketleftBig- 80- 2 λ + λ 2 bracketrightBig = (3- λ )( λ- 10)( λ + 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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E4out - Math 330 Exam 4 Outline Exam 4 for Math 330 will...

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