Lesson C11

C11 Diagonalisation of matrices Quiz: AQA Further Maths, Unit 2

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson C11, Diagonalisation of matrices: 20 multiple choice questions for the AQA Further Maths (7367), Unit 2: Compulsory content, written with Revision Ninja.

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The 20 questions

  1. What does diagonalisation M = UDU^-1 require when the eigenvalues are real?

    • M to have only zero eigenvalues, so that every eigenvector is a zero vector
    • M to be the identity matrix, so that U and D are equal to M
    • M to have trace zero, so that its eigenvalues always add to zero
    • Enough linearly independent eigenvectors, so that U is invertible

  2. In M = UDU^-1, what does D contain?

    • The inverse of M, so that multiplying by D recovers the original matrix
    • The determinant of M on the diagonal, repeated in every position
    • The eigenvectors as its columns, in the same order as the eigenvalues
    • The eigenvalues on the leading diagonal, with zeros elsewhere

  3. What do the columns of U contain in M = UDU^-1?

    • Powers of M
    • Eigenvalues of M
    • Eigenvectors of M
    • Rows of M

  4. For M = [[4, 1], [2, 3]] with eigenvalues 2 and 5, what are the eigenvalues of M^2?

    • 4 and 25
    • 8 and 125
    • 2 and 5
    • 6 and 10

  5. For M = [[3, 1], [0, 2]], what is M^2?

    • [[9, 4], [0, 4]]
    • [[9, 1], [0, 4]]
    • [[9, 5], [0, 4]]
    • [[6, 2], [0, 4]]

  6. For M = [[4, 1], [2, 3]], which eigenvalue belongs to eigenvector (1, 1)?

    • 7
    • 5
    • 2
    • 3

  7. If D = diag(2, 5), what is D^3?

    • diag(2^3, 5)
    • diag(6, 15)
    • diag(8, 15)
    • diag(8, 125)

  8. What happens to M^n as n tends to infinity if M is diagonalisable with eigenvalues 0.5 and -0.3?

    • M^n tends to M
    • M^n tends to the identity
    • M^n diverges
    • M^n tends to the zero matrix

  9. Why is M = [[1, 1], [0, 1]] not diagonalisable?

    • It has only one independent eigenvector for its repeated eigenvalue 1
    • Its determinant is 1, which means the matrix cannot be inverted by U
    • Its eigenvalues are 0 and 2, which are not real numbers in this case
    • Its trace is zero, so the eigenvectors must be orthogonal to each other

  10. Why does (UDU^-1)(UDU^-1) simplify to UD^2U^-1?

    • Because U^-1 U = I, so the middle factors cancel
    • Because U^-1 U = 0, so the middle factors vanish in the product entirely
    • Because D^2 = D, so every power of D equals D itself in this calculation
    • Because D and U commute, so the order of the factors can be swapped freely

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

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

  12. What is the trace of M^2 for M = [[5, 2], [2, 2]]?

    • 7
    • 37
    • 12
    • 41

  13. If a 3 x 3 matrix M has eigenvalues 1, 1 and 2 and is diagonalisable, what is det M?

    • 1
    • 4
    • 2
    • -2

  14. If det U = 3 and det D = 8 for M = UDU^-1, what is det M?

    • 8/3
    • 8
    • 24
    • 3

  15. What is the eigenvalue of M^3 corresponding to eigenvalue 2 of M?

    • 2^(1/3)
    • 5
    • 6
    • 8

  16. If M has eigenvalues 2 and -3, what is det M?

    • -1
    • 6
    • -6
    • 5

  17. If M has eigenvalues 2 and -3, what is the trace of M?

    • -5
    • -1
    • 1
    • 5

  18. What is the eigenvalue of M^4 corresponding to eigenvalue -2 of M?

    • -16
    • 16
    • 8
    • -8

  19. Which matrix has eigenvalue 1 with eigenvector (1, 0) and eigenvalue 3 with eigenvector (1, 2)?

    • [[1, 1], [0, 3]]
    • [[3, 1], [0, 1]]
    • [[1, 2], [0, 3]]
    • [[1, 0], [0, 3]]

  20. What is the determinant of D = diag(2, 5)?

    • 10
    • 3
    • -10
    • 7

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