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1. Show that the matrix A = 0 0 2 2 0 0 0 2 0 is similar to diagonal matrix M3(C) but not in M3(R). 2. Let T ∈ L(V, V ) be a linear transformation such that T r = 1 for some positive integer r. Discuss whether or not there exists a basis of the vector space V consisting of eigenvectors of T. 3. Find the matrix S such that S −1AS is of Jordan canonical form for the following matrix A: A = 3 0 0 0 −1 3 0 0 0 −1 2 0 0 0 −1 3 4. Prove that if A is a nilpotent n × n matrix over C, then the determinant and the trace of A are all equal to zero. 5. Prove that if a real n × n matrix A satisfies the equation x 3 + x 2 + x + 1 = 0, then A is non-singular. Express A−1 by using polynomials of A. 6. Describe how the diagonalization method can be used to solve some real 2 × 2 matrix X such that X2 = −2 9 2 7 . Use this method to solve for X. 7. Let A be an n × n matrix over C and assume that the characteristic polynomial of A has distinct roots. Prove that any two n × n matrices over C which commute with A must commute with each other. (Hint: Consider diagonalization of A). 8. Find an orthonormal basis of R3 which exhibits the principal axes of the quadratic form 2x1x2 + 4x2x3 + 2x3x1.

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