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Question: 1 a the matrix 3 0 01 a 1 3...

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1. (a) The matrix 3 0 01 A 1 3 0 has characteristic polynomial (A - 3)3. Find a basis for the eigenspace E3 and decide whether the matrix A is diagonalizable Show your work (b) Suppose that A and B are n x n matrices and that v 0 is an eigenvector of the mlatrix AB with non-zero eigenvalue λ. Prove that Bv is an eigenvector of BA with eigenvalue λ (dont forget to explain why Bv is a non-zero vector!) 2. Let A - 6 -1 of the corresponding eigenspace E. Show your work. that P-1AP D. Show that the equation PAP D is (a) Find the eigenvalues of A and for each eigenvalue λ, find a basis (b) Write down an invertible matrix P and a diagonal matrix D such satisfied for your matrices P and D.
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