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

The 20 questions

1. What is required to diagonalise an n × n matrix as UDU⁻¹?

   • n zero eigenvalues
   • Symmetric matrix only
   • Zero determinant
   • n independent eigenvectors

2. In the matrix diagonalisation M = UDU⁻¹, what are the diagonal entries of D?

   • Determinants
   • Inverses
   • Eigenvectors
   • Eigenvalues

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. As n tends to infinity, what matrix does Mⁿ approach if all eigenvalues satisfy |λ| < 1?

   • Infinite matrix
   • Inverse matrix
   • Identity matrix
   • Zero matrix

9. Why is the matrix [[1, 1], [0, 1]] not diagonalisable?

   • Insufficient eigenvectors
   • Complex eigenvalues
   • Zero determinant
   • Singular matrix

10. In matrix diagonalisation, why does (UDU⁻¹)(UDU⁻¹) simplify to UD²U⁻¹?

    • U⁻¹U = I
    • DD⁻¹ = I
    • UD = DU
    • UU⁻¹ = D

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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