problems-5-224-S08

# problems-5-224-S08 - M ATH 224 Reading 4.104.11 P ROBLEM S...

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M ATH 224 P ROBLEM S ET 5 D UE : 26 F EBRUARY 2008 Reading. § 4.10–4.11. Problems from the book: 4.9.4, 4.9.6 4.10.2, 4.10.4, 4.10.5, 4.10.12, 4.10.13, 4.10.14 4.11.1, 4.11.2 Additional problems: 1. Compute integraltextintegraltext R ( x + y ) e x 2 - y 2 dA , where R is the region bounded by the lines x - y = 0 , x - y = 2 , x + y = 0 and x + y = 3 . 2. Evaluate integraltextintegraltextintegraltext R xyz | dx dy dz | over the region R between the spheres centered at the origin of radius r = 2 and r = 4 , and above the cone φ = π 3 . 3. Recall that a matrix is orthogonal if and only if AA T = I . Let O ( n ) denote the set of n × n orthogonal matrices. a. Show that O ( n ) is a group. b. Let A O ( n ) and let χ A ( t ) denote its characteristic polynomial. Suppose that α R is a root of χ A ( t ) . Prove that α = ± 1 . c. We have seen in class that a permutation matrix M σ is an orthogonal matrix, for every σ S n . Can you describe all of the roots of the characteristic poly- nomial of M σ ? Your answer should depend on the cycle type of σ . d. We define SO ( n ) = { A O ( n ) | det ( A ) = 1 } . This is the special orthogonal group . Can you give an example of an element of SO ( 2 ) such that none of the roots of the characteristic polynomial are real numbers? As best you can,
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