HW12solutions - Math 221 Assignment 12 Solutions Section 7.4 Problem 4 Diagonalizable The eigenvalues are 0 7 associated with eigenvectors If we let S

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 , 1 3 . If we let S = - 2 1 1 3 , then S - 1 AS = D = 0 0 0 7 . 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 are 3 and 2, with associated eigenvectors 2 1 , 1 1 . If we let S = 2 1 1 1 , then S - 1 AS = D = 3 0 0 2 . Thus A = SDS - 1 and A t = SD t S - 1 = 2 1 1 1 3 t 0 0 2 t 1 - 1 - 1 2 = 2(3 t ) - 2 t 2 t +1 - 2(3 t ) 3 t - 2 t 2 t +1 - 3 t .