---
tag: 0C08BQ3M05
kind: statement
role: theorem
name: Laplace expansion along a row
aliases: [cofactor-expansion]
msc: [15A15]
uses: [0C3CFHDN6R, 0W2QAVJQ1W]
sources:
  - url: https://en.wikipedia.org/wiki/Laplace_expansion
    locator: lead section
---

## Statement

Let $K$ be a commutative ring, let $n \ge 2$, and let $A = (a_{ij})$ be an $n \times n$ matrix over $K$.
Let $C_{ij}$ be the [[cofactors|0XPKR18APA]] of $A$.
Then, for each row $i$,
$$\det A = \sum_{j=1}^{n} a_{ij} C_{ij} = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.$$

## Remarks

The expansion reduces a determinant of order $n$ to $n$ determinants of order $n - 1$.
It is efficient when the chosen row contains many zeros.
There is a similar [[expansion along a column|06AMCJMMVM]].
Replace the entries of row $i$ by those of another row $k$.
Then the sum is $0$, because the new matrix has two equal rows.

## Explanations

### L1 human

To expand a determinant along a row, take each entry of that row.
Multiply it by its cofactor, which is the signed determinant of what is left after deleting its row and column.
Add these products.