Eigenvalues and Eigenvectors · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Directions a matrix only stretches

Mathematics · Linear Algebra · ages 19-20
Name ______________________   Date ____________

  1. The matrix A = [[2, 0], [0, 3]] is diagonal. What are its eigenvalues?

    • 5 and 1
    • 6 and 1
    • 0 and 5
    • 2 and 3

  2. The matrix A = [[2, 0], [0, 3]] is diagonal. What are its eigenvalues?

    • 2 and 3
    • 5 and 1
    • 0 and 5

  3. Take v = (1, 1) and A = [[1, 2], [0, 3]]. Is v an eigenvector of A?

    Circle one:   True   False

  4. Eigenvalues of a square matrix A come from solving which equation?

    • A v = 0
    • det(A) = lambda
    • det(A minus lambda I) = 0

  5. The matrix [[0, minus 1], [1, 0]] rotates the plane by 90 degrees. Does it have any real eigenvalues?

    • Yes: 1 and minus 1
    • No, no real roots
    • Yes: 0

  6. The matrix [[0, -1], [1, 0]] rotates the plane by 90 degrees. Does it have any real eigenvalues?

    • No real eigenvalues
    • Yes: 1 and -1
    • Yes: 0
    • No: its determinant is zero

  7. A = [[4, 1], [2, 3]] has characteristic polynomial lambda squared minus 7 lambda + 10. What is the larger eigenvalue?

    Answer: ______________

  8. A = [[4, 1], [2, 3]] has characteristic polynomial lambda squared minus 7 lambda + 10. What is the larger eigenvalue?

    Answer: ______________

  9. Pat reports the zero vector as an eigenvector. What is wrong?

    • Zero never qualifies
    • Zero qualifies whenever A v = 0
    • Only diagonal matrices reject zero

  10. A 2 by 2 matrix has characteristic polynomial (lambda minus 5)(lambda minus 2). What is the smaller eigenvalue?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Directions a matrix only stretches W1-mt_TVqqaw11qa-s1

  1. 2 and 3 · Diagonal entries are the eigenvalues: 2 and 3.
  2. 2 and 3 · Diagonal entries are the eigenvalues: 2 and 3.
  3. True · Yes: A v = (3, 3) = 3v, so v is an eigenvector with eigenvalue 3.
  4. det(A minus lambda I) = 0 · Nontrivial solutions of (A minus lambda I)v = 0 need a zero determinant.
  5. No, no real roots · Every nonzero vector turns, so no real direction is preserved; the polynomial confirms it.
  6. No real eigenvalues · Every nonzero vector turns, so no real direction is preserved; the polynomial confirms it.
  7. 5 · Factor: (lambda - 5)(lambda - 2) = 0, so the larger root is 5.
  8. 5 · Factor: (lambda minus 5)(lambda minus 2) = 0, so the larger root is 5.
  9. Zero never qualifies · The definition excludes the zero vector, since it stretches trivially under everything.
  10. 2 · (lambda minus 5)(lambda minus 2) = 0 gives roots 5 and 2; smaller is 2.
Worksheet · LightMySky