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7 Eigenvalues and Eigenvectors Online Quiz Questions

Use this free practice quiz with 20 questions to review 7 Eigenvalues and Eigenvectors, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

Let A=[300−2]A=\begin{bmatrix}3&0\\0&-2\end{bmatrix} and v=[01]v=\begin{bmatrix}0\\1\end{bmatrix}. If vv is an eigenvector of AA, what is its eigenvalue?

  1. A

    3

  2. B

    -2

  3. C

    0

  4. D

    2

02
Choose one
1 point

What are the eigenvalues of A=[2014]A=\begin{bmatrix}2&0\\1&4\end{bmatrix}?

  1. A

    2 and 4

  2. B

    1 and 8

  3. C

    -2 and -4

  4. D

    2 and -4

03
Choose one
1 point

Suppose the columns of PP are v1v_1 and v2v_2, where Av1=7v1Av_1=7v_1 and Av2=−v2Av_2=-v_2. Which diagonal matrix DD satisfies A=PDP−1A=PDP^{-1}?

  1. A

    D=[−1007]D=\begin{bmatrix}-1&0\\0&7\end{bmatrix}

  2. B

    D=[7−100]D=\begin{bmatrix}7&-1\\0&0\end{bmatrix}

  3. C

    D=[700−1]D=\begin{bmatrix}7&0\\0&-1\end{bmatrix}

  4. D

    D=[1001]D=\begin{bmatrix}1&0\\0&1\end{bmatrix}

04
Choose one
1 point

A 2×22\times2 matrix has characteristic polynomial p(λ)=(λ−6)2p(\lambda)=(\lambda-6)^2, and its eigenspace for λ=6\lambda=6 has dimension 11. Which conclusion is justified?

  1. A

    It is diagonalizable because it has an eigenvalue.

  2. B

    It is not diagonalizable because the eigenspace dimension is less than the algebraic multiplicity.

  3. C

    It is diagonalizable because its characteristic polynomial has a repeated root.

  4. D

    It cannot have any eigenvectors because its eigenvalue is repeated.

05
Choose all
1 point

Select all statements that are true for the eigenspace associated with an eigenvalue λ\lambda.

  1. A

    The eigenspace EλE_\lambda contains the zero vector.

  2. B

    The zero vector is an eigenvector.

  3. C

    Eλ=ker⁡(A−λI)E_\lambda=\ker(A-\lambda I).

  4. D

    The nonzero vectors in EλE_\lambda are precisely the eigenvectors associated with λ\lambda.

06
Choose all
1 point

For an n×nn\times n matrix over a field F\mathbb F, select all conditions that are equivalent to diagonalizability over F\mathbb F.

  1. A

    The vector space has a basis consisting of eigenvectors of AA.

  2. B

    The total dimension of the eigenspaces is nn.

  3. C

    Every eigenvalue must be distinct.

  4. D

    For every eigenvalue, geometric and algebraic multiplicities agree, and the characteristic polynomial splits over the field.

07
Written response
1 point

Suppose the eigenspace for λ=6\lambda=6 is E6=span⁡{[101],[010]}E_6=\operatorname{span}\left\{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}\right\}. What is the geometric multiplicity of 66?

08
Written response
1 point

What is the standard term for the number of times an eigenvalue occurs as a root of the characteristic polynomial?

09
Fill in the blank
1 point

Complete the definition: The eigenspace associated with λ\lambda is Eλ=E_\lambda=.

10
Fill in the blank
1 point

Complete both identities used in diagonalization: if the columns of PP are appropriately ordered eigenvectors, then AP=AP=, and if PP is invertible, A=A=.

11
Open ended
1 point

Diagonalize A=[1103]A=\begin{bmatrix}1&1\\0&3\end{bmatrix}. Find both eigenvalues, an eigenvector for each eigenvalue, and matrices PP and DD such that A=PDP−1A=PDP^{-1}. Explain why PP is invertible.

12
Choose one
1 point

Which statement correctly describes how the underlying field can affect diagonalizability?

  1. A

    A matrix has the same diagonalizability status over every field.

  2. B

    A real matrix may fail to be diagonalizable over R\mathbb R because of complex eigenvalues, yet be diagonalizable over C\mathbb C.

  3. C

    A real matrix with complex eigenvalues has no eigenvectors over C\mathbb C.

  4. D

    Changing the field changes only the notation, not the available eigenvalues.

13
True or false
1 point

True or false: Eigenvectors corresponding to distinct eigenvalues of a matrix are necessarily linearly independent.

  1. A

    True

  2. B

    False

14
Written response
1 point

What is the standard term for the dimension of the eigenspace associated with an eigenvalue?

15
True or false
1 point

True or false: For an n×nn\times n matrix, defining the characteristic polynomial as det⁡(λI−A)\det(\lambda I-A) or as det⁡(A−λI)\det(A-\lambda I) produces the same eigenvalues.

  1. A

    True

  2. B

    False

16
Written response
1 point

For B=[2102]B=\begin{bmatrix}2&1\\0&2\end{bmatrix}, what is the algebraic multiplicity of the eigenvalue λ=2\lambda=2?

17
Choose one
1 point

Which condition is equivalent to diagonalizability of an n×nn\times n matrix over its underlying field?

  1. A

    It has at least one eigenvalue.

  2. B

    Every eigenvalue is positive.

  3. C

    The sum of the dimensions of its eigenspaces is nn.

  4. D

    Its characteristic polynomial has degree less than nn.

18
Choose one
1 point

A real matrix has a pair of nonreal complex eigenvalues. Which statement is correct?

  1. A

    It is diagonalizable over R\mathbb R because every real matrix has real eigenvalues.

  2. B

    It may fail to be diagonalizable over R\mathbb R but be diagonalizable over C\mathbb C.

  3. C

    It cannot be diagonalizable over either field.

  4. D

    Changing the field changes the matrix entries but not its eigenvalues.

19
Choose one
1 point

If D=[200−1]D=\begin{bmatrix}2&0\\0&-1\end{bmatrix}, what is D3D^3?

  1. A

    [600−3]\begin{bmatrix}6&0\\0&-3\end{bmatrix}

  2. B

    [200−1]\begin{bmatrix}2&0\\0&-1\end{bmatrix}

  3. C

    [800−1]\begin{bmatrix}8&0\\0&-1\end{bmatrix}

  4. D

    [9001]\begin{bmatrix}9&0\\0&1\end{bmatrix}

20
True or false
1 point

True or false: If a nonzero vector vv satisfies Av=−3vAv=-3v, then the transformation represented by AA reverses the direction of vv and triples its length.

  1. A

    True

  2. B

    False