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Contents
- Whole Topic Notes
- 5A/B Eigenvalues and Eigenvectors
- 5C Reducing Matrices to Diagonal Form
- 5D The Cayley–Hamilton Theorem
- Whole Topic Summary Resources (Including Past Paper Questions)
Whole Topic Notes
Full Notes to Accompany Examples
Dr Frost
- Dr Frost Further Pure 2 Matrix Algebra Section
- Dr Frost Further Pure 2 Matrix Algebra Slides
5A/B Eigenvalues and Eigenvectors
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Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 810a: Determine whether a given vector is an eigenvector of a given matrix.
- 810b: Determine the eigenvalue given an eigenvector of a matrix.
- 810c: Determine the characteristic equation of a 2×2 matrix.
- 810d: Determine the eigenvalues of a 2×2 matrix.
- 810e: Determine the eigenvalues of a 3×3 matrix.
- 810f: Determine the eigenvalues and corresponding eigenvectors of a 2×2 matrix.
- 810g: Determine the eigenvalues and corresponding normalised eigenvectors of a 2×2 matrix.
- 810h: Use eigenvectors to state the Cartesian equations of invariant lines of a matrix.
- 810i: Determine the eigenvector of a 3×3 matrix given an eigenvalue.
- 810j: Determine the eigenvalues and corresponding eigenvectors of a 3×3 matrix.
- 810k: Determine an unknown value in a matrix given an eigenvector and corresponding eigenvalue.
- 810: Exam Practice: Eigenvalues, eigenvectors and diagonal form
Worksheets
5C Reducing Matrices to Diagonal Form
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Key Skills
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- 810l: Eigendecompose a diagonalisable matrix into a matrix of eigenvectors \textbf{P} and a diagonal matrix of eigenvalues \textbf{D}.
- 810m: Reduce a 2×2 matrix to diagonal form.
- 810n: Reduce a 3×3 matrix to diagonal form.
- 810o: Diagonalise a symmetric matrix by use of normalised eigenvectors.
- 810p: Use diagonalisation to determine a power of a matrix.
- 810: Exam Practice: Eigenvalues, eigenvectors and diagonal form
5D The Cayley–Hamilton Theorem
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Key Skills
Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost
- 813a: Show that a matrix satisfies its own characteristic equation.
- 813b: Use the Cayley-Hamilton theorem to prove a given matrix equation.
- 813c: Use the Cayley-Hamilton theorem to find the inverse of a 2×2 matrix.
- 813d: Use the Cayley-Hamilton theorem to find the inverse of a 3×3 matrix.
- 813: Exam Practice: The Cayley-Hamilton theorem