---
tag: 09YB3WQG38
kind: statement
role: proposition
name: Permuting rows multiplies the determinant by the sign
msc: [15A15]
uses: [0C3CFHDN6R]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Basic rules
---

## Statement

Let $K$ be a commutative ring, let $A = (a_{ij})$ be an $n \times n$ matrix over $K$, and let $\sigma$ be a permutation of $\{1, \dots, n\}$.
Let $B = (b_{ij})$ be the matrix with $b_{ij} = a_{\sigma(i), j}$.
Then $\det B = \sgn(\sigma) \det A$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

In particular, exchanging two rows changes the sign of the determinant.
For a [[permutation matrix|08SP3DF8NB]] $P$, this gives $\det P = \pm 1$.
The same holds for a permutation of the columns.

## Explanations

### L1 human

When we exchange two rows, the determinant changes its sign.
A general reordering of the rows is a sequence of such exchanges.
So it multiplies the determinant by $1$ or by $-1$, according to the sign of the permutation.