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Unformatted text preview: . (a) Find the 3volume of the parallelepiped deﬁned by the columns of A . 5 (b) Let v be a unit vector in R 4 . The augmented matrix [ A  v ] is 4 × 4 . What is the maximum value det[ A  v ] may take? 6 (3) Let A = ± 54 45 ² . (a) Compute the eigenvalues and the eigenvectors of A . 7 (b) Is the matrix A similar to B = ± 3 1 32 ² ? 8 (4) Let A be the 2 × 2 matrix with eigenvalues 1 2 and 2 , for which ± 2 1 ² and ± 1 2 ² are corresponding eigenvectors. Consider the discrete dynamical system ± x n +1 y n +1 ² = A ± x n y n ² . (a) For the initial value ± x y ² = ± 3 3 ² , ﬁnd the limit of the quotient y n x n as n goes to + ∞ . 9 (b) Find A . 10 (5) Determine whether the following statements are true or false, explaining why. (a) There is a nonzero skewsymmetric matrix A such that I n + A is an orthogonal matrix. (b) For any n × m matrix A , we have im ( A T ) = im ( A T A ) ....
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 Spring '08
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 Linear Algebra, Algebra, Matrices, Orthogonal matrix, Pavlovic A. Yang

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