In this worksheet, we will practice finding the eigenvalues and eigenvectors of a matrix and determining whether a matrix is defective or not.

**Q1: **

For the matrix find the eigenvalues and eigenvectors and determine whether it is defective.

- A , defective
- B , defective
- C , defective
- D , defective
- E , defective

**Q2: **

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

- AThe eigenvalues are 10 with corresponding eigenvector , and 1 with corresponding eigenvector . The matrix is defective.
- BThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . The matrix is not defective.
- CThe eigenvalues are 10 with corresponding eigenvector , and 1 with corresponding eigenvector . The matrix is defective.
- DThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . The matrix is not defective.
- EThe eigenvalues are 1 with corresponding eigenvector , 2 with corresponding eigenvector , and 3 with corresponding eigenvector . The matrix is not defective.

**Q3: **

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

- AThe eigenvalues are 1 with corresponding eigenvector , 3 with corresponding eigenvector , and with corresponding eigenvector . The matrix is not defective.
- BThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is defective.
- C The eigenvalues are 1 with corresponding eigenvector , 3 with corresponding eigenvector , and with corresponding eigenvector . The matrix is defective.
- DThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is not defective.
- EThe eigenvalues are with corresponding eigenvector , with corresponding eigenvector , and 2 with corresponding eigenvector . The matrix is not defective.

**Q4: **

Find the eigenvalues and eigenvectors of the matrix and hence determine whether the matrix is defective.

- AThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
- BThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
- CThe eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
- DThe eigenvalues are 3 with corresponding eigenvector and 6 with corresponding eigenvector .
- E The eigenvalues are with corresponding eigenvector and with corresponding eigenvector .

**Q5: **

- A The eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .
- B The eigenvalues are 2 with corresponding eigenvector and 1 with corresponding eigenvector .
- CThe eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
- DThe eigenvalues are 2 with corresponding eigenvector and 1 with corresponding eigenvector .
- E The eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .

**Q6: **

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

- AA basis for the eigenspace associated with the eigenvalue 2 is , a basis for the eigenspace associated with the eigenvalue 3 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
- BA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore defective.
- CA basis for the eigenspace associated with the eigenvalue 2 is , a basis for the eigenspace associated with the eigenvalue 3 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
- DA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore defective.
- EA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore defective.

**Q7: **

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

- ABases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.

**Q8: **

Find a basis for the eigenspace of each eigenvalue of the matrix and hence determine whether the matrix is defective.

- AA basis for the eigenspace associated with the eigenvalue 7 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
- BA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore defective.
- CA basis for the eigenspace associated with the eigenvalue 7 is , and a basis for the eigenspace associated with the eigenvalue is . The matrix is therefore not defective.
- DA basis for the eigenspace associated with the lone eigenvalue 1 is . The matrix is therefore defective.
- EA basis for the eigenspace associated with the lone eigenvalue 0 is . The matrix is therefore defective.

**Q9: **

- ABases for the eigenspaces associated with the eigenvalues 2 and 6 are , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues 2 and 4 are , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.

**Q10: **

- ABases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore defective.
- BBases for the eigenspaces associated with the eigenvalues 6 and 9 are , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues and are , and respectively. The matrix is therefore defective.
- DBases for the eigenspaces associated with the eigenvalues 6 and 12 are , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues 6 and are , and respectively. The matrix is therefore not defective.

**Q11: **

- ABases for the eigenspaces associated with the eigenvalues and 2 are and respectively. The matrix is therefore defective.
- BBases for the eigenspaces associated with the eigenvalues 1 and 2 are and respectively. The matrix is therefore defective.
- CBases for the eigenspaces associated with the eigenvalues and 2 are and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues 1 and 2 are and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues 1 and are and respectively. The matrix is therefore not defective.

**Q12: **

- ABases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- BBases for the eigenspaces associated with the eigenvalues 4, , and are , , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues 4, , and are , , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues , , and are , , and respectively. The matrix is therefore not defective.

**Q13: **

- ABases for the eigenspaces associated with the eigenvalues 0 and are , respectively. The matrix is therefore defective.
- BBases for the eigenspaces associated with the eigenvalues 0, 13, and are , , and respectively. The matrix is therefore not defective.
- CBases for the eigenspaces associated with the eigenvalues 0, 18, and are , , and respectively. The matrix is therefore not defective.
- DBases for the eigenspaces associated with the eigenvalues 0, 18, and are , , and respectively. The matrix is therefore not defective.
- EBases for the eigenspaces associated with the eigenvalues 0, 13, and are , , and respectively. The matrix is therefore not defective.

**Q14: **

Find the eigenvectors of the matrix

- A
- B
- C
- D
- E

**Q15: **

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 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 1 with corresponding eigenvector .
- D . Its eigenvalues are with corresponding eigenvector and with corresponding eigenvector .
- E . Its eigenvalues are with corresponding eigenvector and 1 with corresponding eigenvector .

**Q16: **

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 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 1 with corresponding eigenvector , with corresponding eigenvector , and with corresponding eigenvector .