201f04fin - B U Department of Mathematics Math 201 Matrix...

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Unformatted text preview: B U Department of Mathematics Math 201 Matrix Theory Fall 2004 Final This archive is a property of Bo˘gazi¸ ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. (a) Show that v = 2 6 6 6 4 7- 1 4 4 3 7 7 7 5 is an eigenvector of A = 2 6 6 6 4 4 2 0 4 0 2- 1 0 0 0 3 3 0 4 0 7 3 7 7 7 5 by finding the corresponding eigenvalue. Solution: Observe that Av = 2 6 6 6 4 42- 6 24 24 3 7 7 7 5 = 6 2 6 6 6 4 7- 1 6 6 3 7 7 7 5 . Hence v is an eigenvector of A corresponding to the eigenvalue 6. (b) Let A and B be two 4 × 4 matrices. Given that B 2 ( AB- B 2 ) B- 1 ( A- B ) 2 = I and det B = 3, find det( A- B ). Solution: I = B 2 ( AB- B 2 ) B- 1 ( A- B ) 2 = B 2 ( A- B ) BB- 1 ( A- B ) 2 = B 2 ( A- B ) 3 . Therefore, det( A- B ) = 3- 2 3 ....
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201f04fin - B U Department of Mathematics Math 201 Matrix...

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