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=ο”βˆ’100βˆ’1.

  • A1,βˆ’1
  • B0,βˆ’1
  • C1, degenerate
  • Dβˆ’1, degenerate

Q2:

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

  • Ayes
  • Bno

Q3:

Fill in the blanks.

Let 𝐴 be an 𝑛×𝑛 matrix. Then trace(𝐴) equals and det(𝐴) equals .

  • Athe product of the eigenvalues of 𝐴; the sum of the eigenvalues of 𝐴
  • Bdet(𝐴); trace(𝐴)
  • Csum of the eigenvalues of 𝐴; negative of the sum of the eigenvalues of 𝐴
  • Dthe sum of the eigenvalues of 𝐴; the product of the eigenvalues of 𝐴

Q4:

Suppose π‘‡βˆˆπΏ(𝑉,𝑉) is a linear operator. Then πœ†βˆˆπΉ is an eigenvalue of 𝑇 if and only if

  • Aπ‘‡βˆ’πœ†πΌ is invertible.
  • B𝑇+πœ†πΌ is not injective.
  • Cπ‘‡βˆ’πœ†πΌ is not injective.
  • Dπ‘‡βˆ’πœ†πΌ is surjective.

Q5:

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

  • A𝐴
  • Bπ΄βˆ—
  • C𝐴
  • D𝐴

Q6:

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

  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=01βˆ’10.

  • A0,2
  • B𝑖,βˆ’π‘–
  • C1+𝑖,1βˆ’π‘–
  • D1,βˆ’1

Q8:

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

  • AThe nullspace of 𝐴 is the zero vector.
  • BThe eigenvalues of 𝐴 are nonzero.
  • Cdet𝐴=1
  • DThe rowspace of 𝐴 is π‘…οŠ©.
  • E𝐴 is nonsingular.

Q9:

What is the characteristic polynomial of the matrix 𝐴=41βˆ’10?

  • Aβˆ’π‘‘+4π‘‘βˆ’1
  • Bβˆ’π‘‘βˆ’4𝑑+1
  • Cπ‘‘βˆ’4𝑑+1
  • Dπ‘‘βˆ’4π‘‘βˆ’1

Q10:

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

  • AYes
  • BNo

Q11:

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

  • Aπœ†=βˆ’1+4π‘–οŠ¨ and 𝑣=3+2π‘–βˆ’1
  • Bπœ†=βˆ’πœ†οŠ¨οŠ§ and 𝑣=βˆ’π‘£οŠ¨οŠ§
  • Cπœ†=βˆ’1βˆ’4π‘–οŠ¨ and 𝑣=ο”βˆ’3βˆ’2π‘–βˆ’1
  • Dπœ†=βˆ’1βˆ’4π‘–οŠ¨ and 𝑣=3βˆ’2π‘–βˆ’1

Q12:

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

  • AYes
  • BNo

Q13:

Let 𝑇 be the linear transformation that reflects all vectors in β„οŠ¨ in the π‘₯-axis. Represent 𝑇 as a matrix and find its eigenvalues and eigenvectors.

  • A𝑇=042βˆ’1. Its eigenvalues are 2 with corresponding eigenvector 23 and 4 with corresponding eigenvector 11.
  • B𝑇=100βˆ’1. Its eigenvalues are βˆ’1 with corresponding eigenvector 01 and 1 with corresponding eigenvector 1βˆ’1.
  • C𝑇=1βˆ’112βˆ’5. Its eigenvalues are βˆ’2 with corresponding eigenvector ο”βˆ’11 and 2 with corresponding eigenvector ο”βˆ’11.
  • D𝑇=100βˆ’1. Its eigenvalues are βˆ’1 with corresponding eigenvector 01 and 1 with corresponding eigenvector 10.
  • E𝑇=042βˆ’1. Its eigenvalues are 2 with corresponding eigenvector 21 and 4 with corresponding eigenvector 11.

Q14:

Let 𝑇 be the linear transformation that reflects all vectors in β„οŠ© in the π‘₯𝑦-plane. Represent 𝑇 as a matrix and find its eigenvalues and eigenvectors.

  • A𝑇=10001000βˆ’1. Its eigenvalues are βˆ’1, with corresponding eigenvector 001, and 1, with corresponding eigenvectors 011 and 100.
  • B𝑇=1000βˆ’1000βˆ’1. Its eigenvalues are βˆ’1, with corresponding eigenvector 001, and 1, with corresponding eigenvectors 010 and 100.
  • C𝑇=100010001. Its only eigenvalue is 1, with corresponding eigenvectors 010 and 111.
  • D𝑇=10001000βˆ’1. Its eigenvalues are βˆ’1, with corresponding eigenvector 001, and 1, with corresponding eigenvectors 010 and 100.
  • E𝑇=100010001. Its only eigenvalue is 1, with corresponding eigenvectors 010 and 100.

Q15:

Find the eigenvectors of the matrix 𝑐000π‘Žβˆ’π‘0π‘π‘Žο₯.

  • A0βˆ’π‘–1ο€οΌβŸ·π‘Žβˆ’π‘–π‘,0𝑖1ο€οΌβŸ·π‘Ž+𝑖𝑏,100ο€οΌβŸ·π‘
  • B0𝑖1ο€οΌβŸ·π‘Ž+𝑖𝑏,0𝑖1ο€οΌβŸ·π‘Žβˆ’π‘–π‘,100ο€οΌβŸ·π‘
  • C0𝑖1ο€οΌβŸ·π‘Žβˆ’π‘–π‘,0𝑖1ο€οΌβŸ·π‘Ž+𝑖𝑏,100ο€οΌβŸ·π‘
  • D0βˆ’π‘–1ο€οΌβŸ·π‘Ž+𝑖𝑏,0𝑖1ο€οΌβŸ·π‘Žβˆ’π‘–π‘,100ο€οΌβŸ·π‘
  • E0𝑖1ο€οΌβŸ·π‘Žβˆ’π‘–π‘,0π‘–βˆ’1ο€οΌβŸ·π‘Ž+𝑖𝑏,ο°ο˜βˆ’100ο€οΌβŸ·π‘

Q16:

Let 𝑇 be the linear transformation that rotates all vectors in β„οŠ¨ counterclockwise through an angle of πœ‹2. Represent 𝑇 as a matrix and find its eigenvalues and eigenvectors.

  • A𝑇=100βˆ’1. Its eigenvalues are βˆ’1 with corresponding eigenvector 01 and 1 with corresponding eigenvector 1βˆ’1.
  • B𝑇=012βˆ’1. Its eigenvalues are 2 with corresponding eigenvector 21 and 4 with corresponding eigenvector 11.
  • C𝑇=0βˆ’110. Its eigenvalues are βˆ’π‘– with corresponding eigenvector ο¬ο”βˆ’π‘–1 and 𝑖 with corresponding eigenvector 𝑖1.
  • D𝑇=0βˆ’110. Its eigenvalues are 𝑖 with corresponding eigenvector 𝑖1 and 1 with corresponding eigenvector 1βˆ’1.
  • E𝑇=100βˆ’1. Its eigenvalues are βˆ’1 with corresponding eigenvector 01 and 1 with corresponding eigenvector 10.

Q17:

Find the eigenvalues and eigenvectors of the matrix 𝑐0000βˆ’π‘0𝑏0ο₯, where 𝑏 and 𝑐 are real numbers.

  • AIts eigenvalues are 𝑐 with corresponding eigenvector 100, βˆ’π‘–π‘ with corresponding eigenvector 0βˆ’π‘–1, and 𝑖𝑏 with corresponding eigenvector 0βˆ’π‘–1.
  • BIts eigenvalues are 𝑐 with corresponding eigenvector 100, βˆ’π‘–π‘ with corresponding eigenvector 0βˆ’π‘–1, and 𝑖𝑏 with corresponding eigenvector 0𝑖1.
  • CIts eigenvalues are 1 with corresponding eigenvector ο˜π‘00, βˆ’π‘– with corresponding eigenvector 0βˆ’π‘–π‘π‘ο₯, and 𝑖 with corresponding eigenvector 0𝑖𝑏𝑏ο₯.
  • DIts eigenvalues are 𝑐 with corresponding eigenvector 100, 𝑖𝑏 with corresponding eigenvector 0βˆ’π‘–1, and βˆ’π‘–π‘ with corresponding eigenvector 0𝑖1.
  • EIts eigenvalues are βˆ’π‘ with corresponding eigenvector ο˜βˆ’100, βˆ’π‘–π‘ with corresponding eigenvector 0βˆ’π‘–1, and 𝑖𝑏 with corresponding eigenvector 0𝑖1.

Q18:

Calculate the eigenvalues of the matrix 𝐴=4βˆ’1βˆ’25.

  • Aβˆ’3 and βˆ’6
  • Bβˆ’92+3√172 and βˆ’92+3√172
  • C3 and 6
  • D92βˆ’3√172 and 92+3√172
  • E92βˆ’βˆš72𝑖 and 92+√72𝑖

Q19:

Calculate the eigenvalues of the matrix 𝐴=11βˆ’120βˆ’1βˆ’223.

  • A1, 3, and 3
  • Bβˆ’3, 1, and 2
  • Cβˆ’1, βˆ’3, and 3
  • Dβˆ’1, 2, and 3
  • E1 and 2

Q20:

Calculate the eigenvalues of the matrix 𝐴=1βˆ’337.

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