Final 2009 - Sample questions for the final Note these do...

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Sample questions for the finalNote these do not cover all of the syllabus for the test and are onlyto be used as a sample.(1) LetAbe a 2×2 matrix with eigenvalues 1 and 4.Let Ker (A-I) =Span{-21}and Ker (A-4I) = Span{3-1}.(a) IsAdiagonalizable? If yes, write out the diagonalization, else explainwhyAis not digaonalizable?(b) Find a diagonal matrixBsuch thatB2=1004.(c) Use parts 1(a) and 1(b) to find a matrixXsuch thatX2=A.(2) LetAbe the matrix2112.(a) Find the eigenvalues and eigenspaces ofA. Write down an orthogonalbasis ofR2consisting of eigenvectors ofA.(b) Find a orthogonal basis ofR2consisting of eigenvectors ofA.(c) LetTdenote the transformation described byA.Write down thematrix ofTwith respect to the new eigenbasis you wrote down in2(a).(d) Explain what the diagonalization ofAdescribes in terms ofT.(3) Solve the following system of differential equations.dx1dt=x1(t)-2x2(t)dx2dt= 2x1(t) +x2(t)Givenx1(0) = 1 andx2(0) =-1. What happens tox1(t), x2(t) ast→ ∞?(4) Find all solutions in C

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