Worksheet: Eigenvalues and Eigenvectors

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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 =  βˆ’ 1 0 0 βˆ’ 1  .

  • A 0 , βˆ’ 1
  • B βˆ’ 1 , degenerate
  • C1, degenerate
  • D 1 , βˆ’ 1

Q2:

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

  • Ano
  • Byes

Q3:

Can a real 3 Γ— 3 matrix which has a nonreal eigenvalue be defective?

  • Ano
  • Byes

Q4:

Fill in the blanks.

Let 𝐴 be an 𝑛 Γ— 𝑛 matrix. Then t r a c e ( 𝐴 ) equals and d e t ( 𝐴 ) equals .

  • Athe product of the eigenvalues of 𝐴 ; the sum of the eigenvalues of 𝐴
  • Bthe sum of the eigenvalues of 𝐴 ; the product of the eigenvalues of 𝐴
  • C d e t ( 𝐴 ) ; t r a c e ( 𝐴 )
  • Dsum of the eigenvalues of 𝐴 ; negative of the sum of the eigenvalues of 𝐴

Q5:

Suppose 𝑇 ∈ 𝐿 ( 𝑉 , 𝑉 ) is a linear operator. Then πœ† ∈ 𝐹 is an eigenvalue of 𝑇 if and only if

  • A 𝑇 + πœ† 𝐼 is not injective.
  • B 𝑇 βˆ’ πœ† 𝐼 is invertible.
  • C 𝑇 βˆ’ πœ† 𝐼 is surjective.
  • D 𝑇 βˆ’ πœ† 𝐼 is not injective.

Q6:

Suppose that πœ† , 𝑣 is an eigenpair of the invertible matrix 𝐴 . Then 1 πœ† , 𝑣 is an eigenpair of which of the following matrices?

  • A 𝐴 βˆ—
  • B 𝐴  
  • C 𝐴 
  • D 𝐴

Q7:

If 𝐴 is a 4 Γ— 4 real matrix, which of the following is true of the matrix 𝐴 𝐴  ?

  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.
  • A2, 3, and 4
  • B1, 2, and 3
  • C2 and 3
  • D1 and 2
  • E1, 3, and 4

Q8:

Find the eigenvalues of 𝐴 =  0 1 βˆ’ 1 0  .

  • A 1 + 𝑖 , 1 βˆ’ 𝑖
  • B 0 , 2
  • C 𝑖 , βˆ’ 𝑖
  • D 1 , βˆ’ 1

Q9:

Suppose 3 Γ— 3 a matrix 𝐴 is row-equivalent to the identity matrix. Which of the following may be false?

  • AThe eigenvalues of 𝐴 are nonzero.
  • B 𝐴 is nonsingular.
  • CThe rowspace of 𝐴 is 𝑅  .
  • D d e t 𝐴 = 1
  • EThe nullspace of 𝐴 is the zero vector.

Q10:

What is the characteristic polynomial of the matrix 𝐴 =  4 1 βˆ’ 1 0  ?

  • A βˆ’ 𝑑 + 4 𝑑 βˆ’ 1 
  • B βˆ’ 𝑑 βˆ’ 4 𝑑 + 1 
  • C 𝑑 βˆ’ 4 𝑑 βˆ’ 1 
  • D 𝑑 βˆ’ 4 𝑑 + 1 

Q11:

Consider a square matrix 𝐴 with eigenvalue πœ† . Is it true that any vector in the nullspace of 𝐴 βˆ’ πœ† 𝐼 is an eigenvector of 𝐴 corresponding to πœ† ?

  • AYes
  • BNo

Q12:

Suppose one eigenvalue of a real matrix 𝐴 is πœ† = βˆ’ 1 + 4 𝑖  and the corresponding eigenvector is 𝑣 =  3 + 2 𝑖 βˆ’ 1  .  Which of the following must also be an eigenpair of 𝐴 ?

  • A πœ† = βˆ’ 1 βˆ’ 4 𝑖  and 𝑣 =  3 βˆ’ 2 𝑖 βˆ’ 1  
  • B πœ† = βˆ’ πœ†   and 𝑣 = βˆ’ 𝑣  
  • C πœ† = βˆ’ 1 βˆ’ 4 𝑖  and 𝑣 =  βˆ’ 3 βˆ’ 2 𝑖 βˆ’ 1  
  • D πœ† = βˆ’ 1 + 4 𝑖  and 𝑣 =  3 + 2 𝑖 βˆ’ 1  

Q13:

Suppose that 𝑣 is an eigenvector of a matrix 𝐴 corresponding to a nonzero eigenvalue πœ† . Is 𝐴 π‘₯ = 𝑣 always solvable?

  • AYes
  • BNo

Q14:

Suppose an π‘š Γ— π‘š matrix 𝐴 is diagonalizable. Then, which of the following does 𝐴 always have?

  • A π‘š linearly independent columns
  • BExactly as many linearly independent eigenvectors as eigenvalues
  • C π‘š distinct eigenvalues
  • D π‘š nonzero eigenvalues
  • E π‘š linearly independent eigenvectors

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