Eigenvalues, eigenvectors and diagonalisation
Analyse 3×3 linear systems, classify consistency and use distinct real eigenvalues to simplify matrix powers.
Exam scope and assessment notes
Mapped to Cambridge International AS & A Level Further Mathematics 9231. These original explanations and self-checks are independent study support, not Cambridge questions, marks or grades. The syllabus assumes prior knowledge of the relevant 9709 Mathematics content; check each component's prerequisites with your teacher. Full-paper assessment and human subject-specialist review remain outstanding.
Before you begin
- Cambridge expects learners to know the whole Cambridge International A Level Mathematics (9709) content. Paper 1 further pure mathematics draws on 9709 Papers 1 and 3. Confirm your exam year and component route with your school.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What makes a Further Mathematics solution convincing?
What you will learn
- Formulate three simultaneous equations as a matrix equation
- Relate consistency to singularity and interpret intersections of planes
- Find real distinct eigenvalues and eigenvectors of 2×2 and 3×3 matrices
- Diagonalise a matrix and use its characteristic equation
Consistency is controlled by the coefficient matrix
Write a system as Ax=b. If A is non-singular, there is one solution. If A is singular, the equations may be inconsistent or dependent; row reduction or geometric interpretation distinguishes these cases.
Three equations in x,y,z describe planes. A unique solution is a common point; dependent equations can share a line or plane, while incompatible equations may have no common point.
Eigenvectors reveal how a matrix acts repeatedly
An eigenvector v satisfies Av=λv, so the matrix only scales that direction. Consequently λⁿ is an eigenvalue of Aⁿ with the same eigenvector. Solve det(A−λI)=0 for eigenvalues, then solve (A−λI)v=0 for each eigenvector.
When a matrix has distinct real eigenvalues and enough independent eigenvectors, put the eigenvectors in the columns of Q and eigenvalues on the diagonal of D. Then A=QDQ⁻¹ and Aⁿ=QDⁿQ⁻¹.
Every square matrix satisfies its characteristic equation. Substituting A for the variable gives a relation among its powers, which can reduce a high power or help find an inverse. The syllabus restricts these examples to 2×2 and 3×3 matrices.
Pause and explain
Which habit best checks your work on eigenvalues, eigenvectors and diagonalisation?
Worked example
Find the eigenvalues and one eigenvector for each eigenvalue of A=[[4,1],[2,3]].
Show the worked solution
- det(A−λI)=(4−λ)(3−λ)−2=λ²−7λ+10=(λ−5)(λ−2).
- For λ=5, (A−5I)v=0 gives v=(1,1).
- For λ=2, (A−2I)v=0 gives v=(1,−2).
Answer Eigenvalues 5 and 2, with corresponding eigenvectors (1,1) and (1,−2).
Try a topic test
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve x+y=3, y+z=5, x+z=4 using a matrix or elimination method.
Give me a hint
Add the first and third equations to find 2x+y+z, or solve pairs by elimination.
Compare my reasoning
- Adding the first and third equations gives 2x+y+z=7; subtract y+z=5 to get 2x=2.
- Then x=1, so x+y=3 gives y=2.
- Finally y+z=5 gives z=3.
(x,y,z)=(1,2,3).
Look for these in your work
- Equations combined correctly
- Variables solved consistently
- All three equations satisfied
Diagonalise A=[[2,0],[0,3]] and use the result to find A⁴.
Give me a hint
The coordinate directions are already eigenvectors.
Compare my reasoning
- The eigenvalues are 2 and 3 with eigenvectors (1,0) and (0,1).
- Choose Q=I and D=A, so A=QDQ⁻¹.
- Raise the diagonal entries to the fourth power.
A⁴=[[16,0],[0,81]].
Look for these in your work
- Eigenpairs identified
- Diagonalisation stated
- Power applied entrywise
Explain why a 3×3 system with singular coefficient matrix may have either no solutions or infinitely many solutions, but not exactly one.
Give me a hint
Consider whether the equations are incompatible or dependent, and how singularity affects invertibility.
Compare my reasoning
- A singular matrix has no inverse, so the usual unique-solution formula cannot apply.
- If the equations contradict one another, the system has no solution.
- If some equations are dependent and consistent, a free variable remains, producing infinitely many solutions.
Singularity means the system is not uniquely solvable: it is inconsistent or has dependent solutions.
Look for these in your work
- Inverse condition explained
- Inconsistent case identified
- Dependent case linked to free variables
Common mistakes
- Treating a singular system as automatically inconsistent
- Confusing eigenvalues with eigenvectors
- Putting eigenvectors in Q but failing to use the same order for the diagonal entries of D
What can you explain now?
What equation defines an eigenvector v with eigenvalue λ?
Compare with the explanation
Av=λv, with v non-zero
The matrix maps that direction to a scalar multiple of itself.
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Make it stick
How do consistency, eigenvectors and diagonalisation help answer different matrix questions?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.