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Worksheet: Eigenvalues of a Matrix

In this worksheet, we will practice determining the eigenvalues of a nonzero square matrix by finding the scalar values that satisfy its characteristic equation.

Q1:

Calculate the eigenvalues of

  • A1, degenerate
  • B 1 , βˆ’ 1
  • C 0 , βˆ’ 1
  • D βˆ’ 1 , degenerate

Q2:

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

  • Ayes
  • Bno

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 .

  • A d e t ( 𝐴 ) ; t r a c e ( 𝐴 )
  • Bsum of the eigenvalues of 𝐴 ; negative of the sum of the eigenvalues of 𝐴
  • Cthe product of the eigenvalues of 𝐴 ; the sum of the eigenvalues of 𝐴
  • Dthe sum of the eigenvalues of 𝐴 ; the product 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 βˆ’ πœ† 𝐼 is invertible.
  • B 𝑇 + πœ† 𝐼 is not injective.
  • 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 𝑇
  • B 𝐴 βˆ—
  • C 𝐴
  • D 𝐴 βˆ’ 1

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