Cofactor Expansion
Compute determinants recursively.
Key idea
Compute determinants recursively.
In depth
det A = Σ aᵢⱼ Cᵢⱼ along any row or column, where Cᵢⱼ = (−1)ⁱ⁺ʲ Mᵢⱼ and Mᵢⱼ is the minor. Pick a row or column with many zeros to save work.
Worked example
Expanding [2 0 1; 3 0 2; 1 4 5] along column 2: only 4 survives, sign (−1)³ = −, minor 2·2 − 1·3 = 1, so det = −4.
Formula
Real-world uses
- Computing determinants for larger systems in physics and stability analysis
Common mistake
Forgetting the checkerboard sign pattern.
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 Cofactor Expansion?
Compute determinants recursively. It is a topic at the expert level in Linear Algebra.
What is the formula for Cofactor Expansion?
The key formula is det A = Σ aᵢⱼ Cᵢⱼ.
Where is Cofactor Expansion used in real life?
Real-world examples: Computing determinants for larger systems in physics and stability analysis.
What should I learn before Cofactor Expansion?
Make sure you are comfortable with Determinant of 3×3, the previous topic in Linear Algebra.
What is a common mistake with Cofactor Expansion?
Forgetting the checkerboard sign pattern.
How can I practise Cofactor Expansion?
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 Cofactor Expansion?
Move on to Properties of Determinants next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.
- Adjugate Matrix – Transpose of the cofactor matrix.