Determinants, Inverses and Linear Systems
Calculating determinants and inverses for two- and three-dimensional matrices, solving simultaneous equations and interpreting solution failure geometrically.
How to study A-level Further Mathematics
Define the objects and conditions, select a representation, carry out exact mathematics, then validate, interpret and communicate the result.
Core concepts
Concept 1
A non-zero determinant identifies an invertible square matrix; a zero determinant signals singularity.
Exam cue: Check determinant and dimensions before attempting an inverse.
Concept 2
An inverse can solve a compatible linear system, but only when the coefficient matrix is non-singular.
Exam cue: Verify a computed inverse by multiplication with the original matrix.
Concept 3
Three simultaneous linear equations may represent planes meeting at one point, along a line, coinciding or having no common point.
Exam cue: Connect algebraic rank or inconsistency with the geometry of the represented planes.
Risk pitfalls and guardrails
Dividing by a zero determinant.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Ignoring equation order when building the coefficient matrix.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Calling every singular system inconsistent when it may have infinitely many solutions.
Guardrail: Do not replace proof with examples, exact reasoning with unverified calculator output, or a valid awarding-body route with an invented mix of options.
Memory anchors
Determinant
The determinant is a scalar that indicates scaling, orientation and whether a square matrix is singular.
Singular Matrix
A singular matrix has determinant zero and no inverse.
Inverse Matrix
The inverse reverses the action of an invertible square matrix.
Linear System
A linear system is a collection of linear equations considered simultaneously.
Geometric Failure
Failure of a unique solution can mean no common intersection or infinitely many common points.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Find det[[3,2],[5,4]].
For A=[[2,1],[3,2]], find A⁻¹.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
What is Pass Harbor?
Completely free exam prep for 247 UK exams.
- Practice questions
- Flashcards
- Study guides
- Mock exams
- No registration
- No paywall
- Start instantly
“No more expensive exam prep. Quality study tools should be accessible to everyone.”