# Worksheet: Eigenvalues and Eigenvectors

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

Q1:

Calculate the eigenvalues of

• A
• B
• C1, degenerate
• D, degenerate

Q2:

Is it possible for a nonzero matrix to have 0 as its only eigenvalue?

• Ayes
• Bno

Q3:

Fill in the blanks.

Let be an matrix. Then ) equals and equals .

• Athe product of the eigenvalues of ; the sum of the eigenvalues of
• B;
• Csum of the eigenvalues of ; negative of the sum of the eigenvalues of
• Dthe sum of the eigenvalues of ; the product of the eigenvalues of

Q4:

Suppose is a linear operator. Then is an eigenvalue of if and only if

• A is invertible.
• B is not injective.
• C is not injective.
• D is surjective.

Q5:

Suppose that , is an eigenpair of the invertible matrix . Then , is an eigenpair of which of the following matrices?

• A
• B
• C
• D

Q6:

If is a real matrix, which of the following is true of the matrix ?

1. It must have nonnegative real eigenvalues.
2. It must have purely imaginary eigenvalues.
3. It must have an orthonormal basis of eigenvectors.
4. It is diagonalizable.
• A1, 3, and 4
• B2, 3, and 4
• C1 and 2
• D2 and 3
• E1, 2, and 3

Q7:

Find the eigenvalues of

• A
• B
• C
• D

Q8:

Suppose a matrix is row-equivalent to the identity matrix. Which of the following may be false?

• AThe nullspace of is the zero vector.
• BThe eigenvalues of are nonzero.
• C
• DThe rowspace of is .
• E is nonsingular.

Q9:

What is the characteristic polynomial of the matrix

• A
• B
• C
• D

Q10:

Consider a square matrix with eigenvalue . Is it true that any vector in the nullspace of is an eigenvector of corresponding to ?

• AYes
• BNo

Q11:

Suppose one eigenvalue of a real matrix is and the corresponding eigenvector is Which of the following must also be an eigenpair of ?

• A and
• B and
• C and
• D and

Q12:

Suppose that is an eigenvector of a matrix corresponding to a nonzero eigenvalue . Is always solvable?

• AYes
• BNo

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 .

Q15:

Find the eigenvectors of the matrix

• A
• B
• C
• D
• E

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