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201y05fin

# 201y05fin - B U Department of Mathematics Math 201 Matrix...

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B U Department of Mathematics Math 201 Matrix Theory Summer 2005 Final Exam 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) [3] A 4 × 4 matrix C is known to have eigenvalues λ 1 = 2 , λ 2 = λ 3 = - 3 and λ 4 = 4.Find det( I + C ), Trace ( I + C ) and det(exp C ) Solution: If C has eigenvalues 2,-3,-3,4 than (I+C) has eigenvalues 3,-2,-2,5 and exp( C ) has eigenvalues e 2 , e - 3 , e - 3 , e 4 then we have det( I + C ) = 3 . ( - 2) . ( - 2) . 5 = 60 Trace ( I + C ) = 3 + ( - 2) + ( - 2) + 5 = 4 det( e C ) = e 2 .e - 3 .e - 3 .e 4 = 1 (b) [3] If K is a skew-symmetric square matrix show that Q = ( I - K )( I + K ) - 1 is an orthogonal matrix. Solution: Compute Q T Q . Q T = [( I + K ) - 1 ] T [ I - K ] T = ( I + K T ) - 1 ( I - K T ) = ( I - K ) - 1 ( I + K ) Q T Q = ( I - K ) - 1 ( I + K )( I - K )( I + K ) - 1 = I since ( I + K )( I - K ) = ( I - K )( I + K ) therefore Q is orthogonal.

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201y05fin - B U Department of Mathematics Math 201 Matrix...

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