Matrices, inverses and geometric transformations
Calculate with small matrices, use determinants and inverses, and interpret 2×2 matrices as transformations of the plane.
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
- Add, subtract and multiply matrices of compatible dimensions
- Find determinants and inverses of non-singular 2×2 and 3×3 matrices
- Use transformation matrices, composition order, area scale factors and invariants
Dimensions, determinants and order control matrix work
A matrix product AB is defined only when the number of columns of A equals the number of rows of B. In general AB and BA are different; for non-singular square matrices, (AB)⁻¹=B⁻¹A⁻¹.
A square matrix is non-singular when its determinant is non-zero. For a 2×2 matrix [[a,b],[c,d]], det=ad−bc and the inverse is (1/det)[[d,−b],[−c,a]]. For a 3×3 matrix, expand the determinant by a row or column and solve Mx=eᵢ for columns of M⁻¹, or use cofactors and the adjugate.
A matrix can transform points and areas
A 2×2 matrix maps a column vector of coordinates to a new point. In AB, B acts first and A acts second. The inverse matrix reverses a non-singular transformation; rotations, reflections, enlargements, stretches and shears have matrix representations.
The area scale factor is |det A|. A determinant of −1 reverses orientation but preserves area. Invariant points satisfy Av=v; an invariant line is mapped to itself, even if its points move along it. Solve (A−I)v=0 for fixed points and substitute a general point on a candidate line to test invariance.
Pause and explain
Which habit best checks your work on matrices, inverses and geometric transformations?
Worked example
Let A=[[2,1],[1,1]]. Find A⁻¹ and state the area scale factor.
Show the worked solution
- det A=2(1)−1(1)=1, so A is non-singular.
- Swap the diagonal entries and negate the other entries: A⁻¹=[[1,−1],[−1,2]].
- The area scale factor is |det A|=1.
Answer A⁻¹=[[1,−1],[−1,2]]; area is preserved.
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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.
Find the inverse of M=[[3,2],[1,1]].
Give me a hint
Calculate the determinant before applying the 2×2 inverse formula.
Compare my reasoning
- det M=3−2=1.
- Swap 3 and 1, then negate 2 and 1.
- Because the determinant is 1, no further scaling is needed.
M⁻¹=[[1,−2],[−1,3]].
Look for these in your work
- Determinant correct
- Adjugate entries correct
- Inverse verified by determinant condition
A=[[1,1],[0,1]] and B=[[0,−1],[1,0]]. Find AB and state which transformation acts first when AB is applied to a point.
Give me a hint
Multiply rows of A by columns of B; the rightmost matrix acts first.
Compare my reasoning
- AB=[[1,−1],[1,0]].
- For a column vector v, ABv=A(Bv).
- So B acts first, then A.
AB=[[1,−1],[1,0]]; B acts first, then A.
Look for these in your work
- Matrix product calculated
- Order interpreted from right to left
- Both matrices named in order
A non-singular matrix T fixes every point on the line y=x. What does that tell you about the line, and what equation can you use to find other invariant points?
Give me a hint
A fixed line is invariant; a fixed point satisfies Tv=v.
Compare my reasoning
- The line y=x is invariant because its points remain on the line.
- A point is fixed when T(x,y)ᵀ=(x,y)ᵀ.
- Rearrange (T−I)v=0 and solve together with the required conditions.
The line is invariant; fixed points solve (T−I)v=0.
Look for these in your work
- Invariant line distinguished from fixed points
- Fixed-point equation correct
- Linear system stated
Common mistakes
- Multiplying matrices in the wrong order
- Treating determinant zero as an ordinary invertible transformation
- Using det A rather than |det A| for area scale
What can you explain now?
If a 2D transformation has determinant −3, what happens to area and orientation?
Compare with the explanation
Area is multiplied by 3 and orientation is reversed
Area uses the absolute determinant; the negative sign records orientation reversal.
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Make it stick
What do determinant, inverse and product order tell you about a matrix transformation?
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.