Linear algebra multiple choice

[underacheiver]

Homework Statement

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

Homework Equations

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