---
tag: 0K4VQB7HWR
kind: statement
role: proposition
name: Determinant of a matrix with two equal rows
msc: [15A15]
uses: [0C3CFHDN6R]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Basic rules
---

## Statement

Let $K$ be a commutative ring and let $A$ be an $n \times n$ matrix over $K$.
If two different rows of $A$ are equal, then $\det A = 0$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This is property 2 of the definition of the determinant.
The same holds for two equal columns.
Over a field, more generally, $\det A = 0$ when the rows of $A$ are linearly dependent.

## Explanations

### L1 human

Two equal rows mean that the rows span less than the full space.
Then the box spanned by the rows is flat, and its volume is zero.
So the determinant is zero.