---
tag: 04CPJ6AF00
kind: statement
role: theorem
name: Determinant is multiplicative
msc: [15A15]
uses: [0C3CFHDN6R, 076JGP6R3T]
sources:
  - url: https://en.wikipedia.org/wiki/Determinant
    locator: Basic rules
---

## Statement

Let $K$ be a commutative ring and let $A$ and $B$ be $n \times n$ matrices over $K$.
Then
$$\det(AB) = \det A \cdot \det B,$$
where $AB$ is the [[matrix product|032Q9FB8MT]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

Thus $\det$ is a group homomorphism from the [[general linear group|0FCHE902D7]] $GL_n(F)$ to $F^\times$, over a field $F$.
It follows that $\det(AB) = \det(BA)$, although $AB \ne BA$ in general.
Similar matrices have the same determinant, because $\det(P^{-1} A P) = \det A$.

## Explanations

### L1 human

If one matrix scales volume by $\det A$ and another by $\det B$, then doing both scales volume by $\det A \cdot \det B$.
This is the meaning of the product rule.
It is one of the most useful properties of the determinant.