Cramer’s Rule
Solve linear systems using determinants.
Key idea
Solve linear systems using determinants.
In depth
For a system Ax = b with det A ≠ 0, each unknown equals a ratio of determinants: xᵢ = det(Aᵢ)/det(A), where Aᵢ has column i replaced by b. Neat for small systems.
Worked example
x + y = 3, x − y = 1: det A = −2; replacing columns gives −4 and −2, so x = −4/−2 = 2 and y = −2/−2 = 1.
Formula
Real-world uses
- Solving small systems of equations in physics and engineering when a formula is needed
Common mistake
Using it when det A = 0; there is no unique solution.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Cramer’s Rule?
Solve linear systems using determinants. It is a topic at the expert level in Algebra.
What is the formula for Cramer’s Rule?
The key formula is xᵢ = det(Aᵢ)/det(A).
Where is Cramer’s Rule used in real life?
Real-world examples: Solving small systems of equations in physics and engineering when a formula is needed.
What should I learn before Cramer’s Rule?
Make sure you are comfortable with Matrices (Algebra Intro), the previous topic in Algebra.
What is a common mistake with Cramer’s Rule?
Using it when det A = 0; there is no unique solution.
How can I practise Cramer’s Rule?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Cramer’s Rule?
Move on to Conic Sections: Overview next.
Related expert topics in Algebra
- Linear Programming – Optimise a linear objective over a feasible polygon; maxima occur at vertices.
- Rational Root Theorem – Possible rational roots are ± factors of constant over factors of leading coefficient.
- Fundamental Theorem of Algebra – Every degree-n polynomial has exactly n complex roots counted with multiplicity.
- Binomial Theorem – Expand (a+b)ⁿ with binomial coefficients.
- Partial Fractions – Decompose a rational function into simpler fractions.
- Mathematical Induction – Prove statements for all n by base case and inductive step.