The Further Pure 2 treatment goes past inversion into diagonalisation, which turns awkward repeated multiplication into a routine calculation.
Syllabus points covered
- Find eigenvalues and eigenvectors of 2×2 and 3×3 matrices
- Express a matrix in diagonalised form where possible
- Use diagonalisation to find powers of a matrix
- Use the Cayley-Hamilton theorem and understand its consequences
Common mistakes
- Ordering the eigenvectors in P differently from the eigenvalues in D.
- Attempting diagonalisation for a matrix without enough independent eigenvectors.
- Making arithmetic errors in the characteristic equation — check it by summing eigenvalues against the trace.
- Computing Mⁿ as PDⁿP rather than PDⁿP⁻¹.
Worksheets
Use these as soon as you've learned the topic and need to practise it. Start with the worksheet, check yourself against the answers, and only then look at the step-by-step solutions.
- Practice worksheetComing soon
- Worksheet answersComing soon
- Step-by-step solutionsComing soon
Topical past papers
Move on to these once you've worked through the worksheets and are ready for harder, exam-style questions. Real past-paper questions on this topic, with the official mark scheme and full worked solutions.
- Topical questions
- Mark scheme
- Step-by-step worked solutionsComing soon
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