---
tag: 01EXSMK864
kind: statement
role: proposition
name: Determinant of a scalar multiple
msc: [15A15]
uses: [0C3CFHDN6R, 0K28YXMS7Q]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Basic rules
---

## Statement

Let $K$ be a commutative ring, let $A$ be an $n \times n$ matrix over $K$, and let $c \in K$.
Then
$$\det(cA) = c^n \det A,$$
where $cA$ is the [[scalar multiple|09V6DDACXR]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Each of the $n$ rows is multiplied by $c$, and each row contributes one factor $c$.
For example, $\det(-A) = (-1)^n \det A$.
So an $n \times n$ real [[skew-symmetric matrix|0PBY4E9GQF]] with $n$ odd has determinant $0$.

## Explanations

### L1 human

Scaling a matrix by $c$ stretches every edge of a box by $c$.
In $n$ dimensions, this multiplies the volume by $c^n$.
So doubling a $3 \times 3$ matrix multiplies its determinant by $8$.