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

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1. (a) The matrix 3 0 0 A 3 0 has characteristic polynomial (X - 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 f 0 is an eigenvector of the matrix 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!)
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