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A-LevelFurther MathematicsMatrices9231/12 · May/June 2019 · Q910 Marks

It is given that e is an eigenvector of the matrix A, with corresponding eigenvalue λ.

  1. (i)Show that e is an eigenvector of A², with corresponding eigenvalue λ².[2]
  2. (ii)The matrices A and B are given by A= (n 1 3; 0 2n 0; 0 0 3n) and B = (A + nI)², where I is the 3 × 3 identity matrix and n is a non-zero integer. Find, in terms of n, a non-singular matrix P and a diagonal matrix D such that B = PDP⁻¹.[8]

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

  1. [2] AG
  2. [1] Eigenvalues of A are n,2n and 3n.
  3. [2] Uses vector product (or equations) to find corresponding eigenvectors.
  4. [1]
  5. [1]
  6. [1] Eigenvalues of A+nI are 2n,3n and 4n
  7. [2] Or correctly matched permutations of columns.

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About this question

Topic
Matrices, A-Level Further Mathematics (syllabus 9231). Structured question, 10 marks.
Source
A-Level Further Mathematics, May/June 2019 Paper 1 Variant 2, question 9.

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