# Determinant

A single number computed from a square matrix. For [[a,b],[c,d]] it is ad - bc. It tells you how much the matrix scales areas.

Maths (university) · Part 09: Determinants, inverses and vector spaces · Card 1 of 9 · Last updated 2026-10-19

## Analogy

A stretch factor: how many times bigger a shape becomes after the matrix acts on it.

## Example

**Matrix [[3,1],[2,4]]**

det = 3(4) - 1(2).
= 12 - 2.
= 10.
A unit square becomes area 10.

## In practice

A zero determinant means the matrix squashes space flat and cannot be undone.

## Practice

### Flashcards

- **What is the determinant of [[a,b],[c,d]]?** ad - bc.
- **What is the determinant of [[3,1],[2,4]]?** 10.
- **What does a zero determinant mean?** The matrix has no inverse.

## Questions people ask

### What does a negative determinant mean?

The matrix flips space over, like looking in a mirror, as well as scaling it. The size of the number is still the area scale factor. A determinant of -10 means areas grow tenfold and the orientation is reversed. Swapping two rows of a matrix has this same effect on its determinant.

### What does a determinant of zero tell me?

The matrix squashes the plane onto a line or a point, so some area is lost and cannot be recovered. That means the matrix has no inverse, and a system with that matrix does not have exactly one solution. Instead, it has none or infinitely many.

## Related concepts

- [Inverse matrix](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-3)
- [Cofactor expansion](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-2)
- [Trace and determinant check](https://studyvisually.com/learn/science/maths-university/maths-university-part-10/maths-university-part-10-concept-4)

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