---
tag: 0FS6B46E5T
kind: statement
role: proposition
name: Determinant of a matrix with a zero row
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 one row of $A$ is zero, then $\det A = 0$, where $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

This follows from linearity in that row: the zero row is $0$ times itself, so $\det A = 0 \cdot \det A = 0$.
The same holds for a zero column.

## Explanations

### L1 human

A zero row means that one edge of the box spanned by the rows has length zero.
Such a box has no volume.
So the determinant is zero.