HW12solutions - Math 221, Assignment 12 Solutions Section...

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Math 221, Assignment 12 Solutions Section 7.4 Problem 4 . Diagonalizable. The eigenvalues are 0, 7 associated with eigenvectors ± - 2 1 ² , ± 1 3 ² . If we let S = ± - 2 1 1 3 ² , then S - 1 AS = D = ± 0 0 0 7 ² . Problem 10 . Not diagonalizable. There is only one eigenvalue, 1, with a one-dimentional eigenspace. Problem 12 . Diagonalizable. The eigenvalues are 2, 1, 1, with associated eigenvectors, 1 0 0 , 0 1 0 , - 1 0 1 . If we let S = 1 0 - 1 0 1 0 0 0 1 , then S - 1 AS = D = 2 0 0 0 1 0 0 0 1 . Problem 26 . The eigenvalues are 1, 2, 1 and the matrix is diagonalizable if (and only if) the eigenspace E 1 is two-dimensional. Now E 1 = ker( A - I ) = ker 0 a b 0 1 c 0 0 0 = 0 1 c 0 0 b - ac 0 0 0 is two-dimensional if (and only if) b - ac = 0. Thus the matrix is diagonalizable if and only if b - ac = 0. Problem 32 . The eigenvalues of A = ± 4 - 2 1 1 ² are 3 and 2, with associated eigenvectors ± 2 1 ² , ± 1 1 ² . If we let S = ± 2 1 1 1 ²
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This homework help was uploaded on 02/23/2008 for the course MATH 2210 taught by Professor Pantano during the Fall '05 term at Cornell University (Engineering School).

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HW12solutions - Math 221, Assignment 12 Solutions Section...

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