C10 Eigenvalues and eigenvectors Quiz: AQA Further Maths, Unit 2

The 20 questions

1. What is the characteristic equation of a matrix M?

   • det(M - lambda I) = 0
   • det(M) = lambda
   • det(M + lambda I) = 1
   • M - lambda I = I

2. What are the eigenvalues of [[2, 0], [0, 5]]?

   • 2 and 0
   • 0 and 7
   • 1 and 5
   • 2 and 5

3. Which statement describes an eigenvector?

   • A vector that is always reflected by the transformation
   • A vector that is only scaled, not rotated, by the transformation
   • A vector that always has length 1
   • A vector that is mapped to the zero vector

4. What are the eigenvalues of [[4, 1], [2, 3]]?

   • 1 and 6
   • 3 and 4
   • -2 and -5
   • 2 and 5

5. Which vector is an eigenvector of [[4, 1], [2, 3]] for eigenvalue 5?

   • (1, -1)
   • (2, 1)
   • (1, 2)
   • (1, 1)

6. Which vector is an eigenvector of [[4, 1], [2, 3]] for eigenvalue 2?

   • (1, -2)
   • (2, -1)
   • (1, 2)
   • (2, 1)

7. What are the eigenvalues of [[2, 1, 0], [0, 2, 1], [0, 0, 5]]?

   • 2, 2 and 5
   • 2 and 5 only
   • 1, 2 and 5
   • 2, 5 and 7

8. What is the characteristic polynomial of [[1, 2], [3, 4]]?

   • lambda^2 - 5 lambda - 2
   • lambda^2 - 3 lambda - 2
   • lambda^2 - 5 lambda + 2
   • lambda^2 + 5 lambda - 2

9. What is the product of the eigenvalues of [[4, 1], [2, 3]]?

   • 12
   • 7
   • 10
   • -10

10. A matrix has eigenvalue 3 with eigenvector (1, 1). What is the eigenvalue of M^2 for that eigenvector?

    • 9
    • 6
    • 3
    • -1

11. For which real k does [[k, 1], [1, k]] have 0 as an eigenvalue?

    • k = 2 only, since the matrix then has trace 4 and determinant 3
    • No real k, since a 2 x 2 matrix can never have eigenvalue 0
    • k = 1 or k = -1
    • k = 0 only, since the matrix then has equal diagonal entries

12. What are the eigenvalues of the lower triangular matrix [[2, 0, 0], [1, 3, 0], [0, 1, 4]]?

    • 2, 4 and 6
    • 2, 3 and 4
    • 2, 3 and 7
    • 1, 3 and 4

13. Which matrix has eigenvalues 1 and 2?

    • [[2, 0], [0, 2]]
    • [[1, 0], [0, 2]]
    • [[1, 2], [0, 1]]
    • [[1, 1], [1, 1]]

14. What does an eigenvalue of -1 with eigenvector v mean geometrically?

    • v is unchanged in length and direction, so the transformation leaves it fixed
    • v is mapped to the zero vector, so the eigenvalue shows the vector collapses
    • v is rotated through 90 degrees, so the eigenvector direction changes
    • v is mapped to -v, so it is reversed along its own line

15. What are the eigenvalues of [[0, 1], [-2, -3]]?

    • -1 and 2
    • 1 and 2
    • -3 and 2
    • -1 and -2

16. What are the eigenvalues of [[5, 2], [2, 2]]?

    • 1 and 6
    • 2 and 5
    • 3 and 4
    • -1 and -6

17. What are the eigenvalues of [[3, 0], [0, -2]]?

    • 3 and 2
    • 3 and -2
    • 1 and -2
    • -3 and 2

18. Which quadratic is the characteristic equation of [[2, 1], [0, 2]]?

    • lambda^2 - 2 lambda + 4 = 0
    • lambda^2 + 4 lambda + 4 = 0
    • lambda^2 - 4 lambda + 4 = 0
    • lambda^2 - 4 lambda - 4 = 0

19. What are the eigenvalues of [[0, 1], [1, 0]]?

    • -1 and -2
    • 0 and 1
    • 1 and -1
    • 1 and 2

20. What are the eigenvalues of [[5, -1], [2, 2]]?

    • 1 and 6
    • 3 and 4
    • -3 and -4
    • 2 and 5

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