Final 2009 Solutions - Partial solutions to Sample...

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Partial solutions to Sample questions for the finalPlease use these only as a reference, there might be typos and errors.(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?Hint. Since
(b) Find a diagonal matrixBsuch thatB2=1004.
(c) Use parts 1(a) and 1(b) to find a matrixXsuch thatX2=A.
2. Substitute to findX.(2) LetAbe the matrix2112.(a) Find the eigenvalues and eigenspaces 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.

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