# Cofactor expansion

Compute a larger determinant by going along a row, multiplying each entry by the determinant of the smaller matrix left after crossing out its row and column. Signs alternate + - +.

Maths (university) · Part 09: Determinants, inverses and vector spaces · Card 2 of 9 · Last updated 2026-10-19

## Analogy

Breaking a big job into smaller jobs of the same kind, until each is easy.

## Example

**[[1,2,3],[0,1,4],[5,6,0]]**

Row 1: 1(1x0 - 4x6) = -24.
-2(0x0 - 4x5) = +40.
+3(0x6 - 1x5) = -15.
Total -24 + 40 - 15 = 1.

## In practice

The standard hand method for 3 by 3 determinants.

## Practice

### Flashcards

- **What do the signs follow in cofactor expansion?** A checkerboard pattern starting with plus.

## Questions people ask

### Which row should I expand along?

Choose the row or column with the most zeros, since each zero removes a whole smaller determinant from the work. Any row or column gives the same answer, as long as you apply the alternating plus and minus signs correctly.

### How do the signs in cofactor expansion work?

The signs follow a checkerboard pattern starting with plus in the top left corner. Along the first row, they go plus, minus, plus. Along the second row they go minus, plus, minus. Forgetting the signs is the most common mistake.

## Related concepts

- [Determinant](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-1)
- [Inverse matrix](https://studyvisually.com/learn/science/maths-university/maths-university-part-9/maths-university-part-9-concept-3)
- [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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