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Unformatted text preview: Name: 1 Math 20 Fall 2010 Final Exam (1) (10 points) For a system of linear equations in 2 variables x and y , how many solutions could it have? For each case, give an example of a system of linear equations in two variables with that many solutions. 2 (2) (14 points) Find the inverse of the following matrix using row reduction. Make sure you clearly label the inverse so I know you’ve found it. Using your row reduction steps, find the determinant of the matrix. DO NOT use cofactor expansion, except to check your answer, if you wish. 0 0 2 0 0 1 0 0 3 3 0 0 0 0 0 1 Name: 3 (3) (12 points) Let ~v = ( 1 , 2 , 1 , 0) and ~w = (3 , 1 , 1 , 2). (a) Calculate ~v · ~w . (b) Calculate the lengths of each vector. (c) Verify the CauchySchwarz inequality for these vectors. 4 (4) (8 points) Let A be a given n × n matrix. Show that the set of all vectors ~x satisfying A~x = ~ 0 is a vector space. (5) (8 points) As you know, we find eigenvectors of A with eigenvalue λ by solving the matrix equation ( A λI ) ~v = ~ 0, where I is the identity matrix. However, the definition of eigenvector was that A~v = λ~v (for nonzero ~v ). Show that any ~v satisfying ( A λI ) ~v = ~ 0 also satisfies A~v = λ~v . (In fact, the two equations are equivalent, so all of the eigenvectors can be found by solving the first equation.) Name: 5 (6) Let A be the matrix A =  1 4 4 3 1 3 . (a) (6 points) Find the eigenvalues of A along with their algebraic multiplicities. (b) (8 points) Find all eigenvectors of A along with the geometric multiplicities of the eigenvalues. 6 (c) (6 points)Is A diagonalizable? If so, find matrices P and D such that P 1 AP = D . If not, explain how you know that it’s not....
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This note was uploaded on 03/27/2012 for the course MATH 20 taught by Professor Rachelepstein during the Fall '11 term at Harvard.
 Fall '11
 RachelEpstein
 Linear Equations, Equations

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