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Homework Help: Linear algebra multiple choice

  1. Jan 25, 2010 #1
    1. The problem statement, all variables and given/known data
    1. If A is a real symmetric matrix, then there is a diagonal matrix D and an orthogonal matrix P so that D = P T AP.
    a. True
    b. False

    2. Given that λi and λj are distinct eigenvalues of the real symmetric matrix A and that v1 and v2 are the respective eigenvectors associates with these values, then v1 and v2 are orthogonal.
    a. True
    b. False

    3.If T(θ) is a rotation of the Euclidean plane 2 counterclockwise through an angle θ, then T can be represented by an orthogonal matrix P whose eigenvalues are λ1 = 1 and λ2 = -1.
    a. True
    b. False

    4. If A and B represent the same linear operator T: U → U, then they have the same eigenvalues.
    a. True
    b. False

    5. If A and B represent the same linear operator T: U → U, then they have the same eigenvectors.
    a. True
    b. False

    6. If A and B have the same eigenvalues, then they are similar matrices.
    a. True
    b. False

    7. Which of the following statements is not true?
    a. Similar matrices A and B have exactly the same determinant.
    b. Similar matrices A and B have exactly the same eigenvalues.
    c. Similar matrices A and B have the same characteristic polynomial.
    d. Similar matrices A and B have exactly the same eigenvectors.
    e. none of the above

    8. Let the n × n matrix A have eigenvalues λ1, λ2 ... λn (not necessarily distinct). Then det(A) = λ1λ2 ... λn.
    a. True
    b. False

    9. Every real matrix A with eigenvalues as in problem 8 is similar to the diagonal matrix D = diag [λ1, λ2, ... λn].
    a. True
    b. False

    10. Eigenvectors corresponding to distinct eigenvalues for any n × n matrix A are always linearly independent.
    a. True
    b. False


    2. Relevant equations



    3. The attempt at a solution
    1. b
    2. a
    3. a
    4. a
    5. b
    6. b
    7. d
    8. a
    9. b
    10. a
     
  2. jcsd
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