---
tag: 0PPG793642
kind: statement
role: proposition
name: Determinant is multilinear in the rows
msc: [15A15]
uses: [0C3CFHDN6R]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Basic rules
---

## Statement

Let $K$ be a commutative ring, let $n \ge 1$, and let $1 \le i \le n$.
Let $A$, $B$, $C$ be $n \times n$ matrices over $K$ that agree outside row $i$.
Let $a$, $b$, $c$ be row $i$ of $A$, $B$, $C$, and assume $c = sa + tb$ with $s, t \in K$.
Then
$$\det C = s \det A + t \det B,$$
where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 1 of the definition of the determinant.
The determinant is not additive in the whole matrix: in general $\det(A + B) \ne \det A + \det B$.
By $\det A^T = \det A$, the determinant is also multilinear in the columns.

## Explanations

### L1 human

If we fix all rows but one, then the determinant is a linear function of the remaining row.
So a sum in one row splits the determinant into a sum of two determinants.
A factor in one row comes out of the determinant.