(adsbygoogle = window.adsbygoogle || []).push({}); 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

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# Homework Help: Linear algebra multiple choice

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