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
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
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 =
. 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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