# Worksheet: Eigenvalues and Eigenvectors

In this worksheet, we will practice finding the eigenvalues of a matrix and the corresponding eigenvectors of a matrix.

**Q13: **

Let be the linear transformation that reflects all vectors in in the -axis. Represent as a matrix and find its eigenvalues and eigenvectors.

- A. Its eigenvalues are 2 with corresponding eigenvector and 4 with corresponding eigenvector .
- B. Its eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
- C. Its eigenvalues are with corresponding eigenvector and 2 with corresponding eigenvector .
- D. Its eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
- E. Its eigenvalues are 2 with corresponding eigenvector and 4 with corresponding eigenvector .

**Q14: **

Let be the linear transformation that reflects all vectors in in the -plane. Represent as a matrix and find its eigenvalues and eigenvectors.

- A. Its eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .
- B. Its eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .
- C. Its only eigenvalue is 1, with corresponding eigenvectors and .
- D. Its eigenvalues are , with corresponding eigenvector , and 1, with corresponding eigenvectors and .
- E. Its only eigenvalue is 1, with corresponding eigenvectors and .

**Q16: **

Let be the linear transformation that rotates all vectors in counterclockwise through an angle of . Represent as a matrix and find its eigenvalues and eigenvectors.

- A. Its eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
- B. Its eigenvalues are 2 with corresponding eigenvector and 4 with corresponding eigenvector .
- C. Its eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
- D. Its eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
- E. Its eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .

**Q17: **

Find the eigenvalues and eigenvectors of the matrix where and are real numbers.

- AIts eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
- BIts eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
- CIts eigenvalues are 1 with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
- DIts eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .
- EIts eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .

**Q18: **

Calculate the eigenvalues of the matrix

- A and
- B and
- C3 and 6
- D and
- E and

**Q19: **

Calculate the eigenvalues of the matrix

- A1, 3, and 3
- B, 1, and 2
- C, , and 3
- D, 2, and 3
- E1 and 2

**Q20: **

Calculate the eigenvalues of the matrix