Determinant of 3×3
Expand along a row or use the rule of Sarrus.
Key idea
Expand along a row or use the rule of Sarrus.
In depth
Compute by cofactor expansion along a row, or by the rule of Sarrus (add diagonals going down, subtract those going up). The result is the signed volume scaling factor.
Worked example
det [1 2 3; 0 1 4; 5 6 0] = 1(0 − 24) − 2(0 − 20) + 3(0 − 5) = 1.
Real-world uses
- Calculating the volume of a 3D object spanned by three vectors in engineering
Common mistake
Applying Sarrus to 4 × 4 matrices, where it does not work.
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 Determinant of 3×3?
Expand along a row or use the rule of Sarrus. It is a topic at the advanced level in Linear Algebra.
Where is Determinant of 3×3 used in real life?
Real-world examples: Calculating the volume of a 3D object spanned by three vectors in engineering.
What should I learn before Determinant of 3×3?
Make sure you are comfortable with Determinant of 2×2, the previous topic in Linear Algebra.
What is a common mistake with Determinant of 3×3?
Applying Sarrus to 4 × 4 matrices, where it does not work.
How can I practise Determinant of 3×3?
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 Determinant of 3×3?
Move on to Cofactor Expansion next.
Related advanced topics in Linear Algebra
- Dot Product – Sum of component products.
- Angle Between Vectors – Cosine from the dot product.
- Orthogonality – Vectors are orthogonal when their dot product is 0.
- Projection – Component of one vector along another.
- Cross Product – A vector perpendicular to two vectors in 3D.
- Lines and Planes in Vector Form – Parametric and normal equations.