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

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Math 221, Assignment 12 SolutionsSection 7.4Problem 4.Diagonalizable. The eigenvalues are 0, 7 associated with eigenvectors-21,13.If we letS=-2113, thenS-1AS=D=0007.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,100,010,-101. If we letS=10-1010001, thenS-1AS=D=200010001.Problem 26.The eigenvalues are 1, 2, 1 and the matrix is diagonalizable if (and only if) theeigenspaceE1is two-dimensional. NowE1= ker(A-I) = ker0ab01c000=01c00b-ac000is two-dimensional if (and only if)b-ac= 0.Thus the matrix is diagonalizable if andonly ifb-ac= 0.Problem 32.The eigenvalues ofA=4-211are 3 and 2, with associated eigenvectors21,11.If we letS=2111, thenS-1AS=D=3002.ThusA=SDS-1andAt=SDtS-1=21113t002t1-1-12=2(3t)-2t2t+1-2(3t)3t-2t2t+1-3t.

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