mid2sol - Math 115a Midterm 2 Lecture 1 Fall 2009 Name:...

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Unformatted text preview: Math 115a Midterm 2 Lecture 1 Fall 2009 Name: Instructions: • There are 5 problems. Make sure you are not missing any pages. • Unless stated otherwise, you may use without proof anything proven in the sections of the book covered by this test. • You may only cite an exercise from the book if it was assigned as homework. • You must prove your answers (OF COURSE!). Question Points Score 1 15 2 15 3 10 4 10 5 10 Total: 60 1. (15 points) (a) (8 points) Find the eigenvalues of the following matrix, de- termine whether it is diagonalizable, and find one eigenvector. 5 4 4 5 (b) (7 points) Find the eigenvalues of the following matrix and determine whether it is diagonalizable. 1 2 3 0 2 2 0 0 1 Solution (a): f A ( t ) = det 5- t 4 4 5- t = (5- t ) 2- 16 = ( t- 9)( t- 1) , so the eigenvalues are 9 and 1. Suppose ( A- I 2 ) v = 0. Then- 4 v 1 + 4 v 2 = 0, so E 9 = span { (1 , 1) } . Similarly, E 1 = span { (- 1 , 1) } . The matrix is diagonalizable because it has two distinct eigenvalues. Solution (b): f A ( t ) = det 1- t 2 3 2- t 2 1- t = (1- t ) 2 (2- t ) , so the eigenvalues are 1 (with multiplicity 2) and 2 (with multiplicity 1). Since theso the eigenvalues are 1 (with multiplicity 2) and 2 (with multiplicity 1)....
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This note was uploaded on 02/22/2010 for the course MATH Math 115 taught by Professor Taft during the Fall '06 term at UCLA.

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mid2sol - Math 115a Midterm 2 Lecture 1 Fall 2009 Name:...

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