---
tag: 0G24VQ1088
kind: statement
role: theorem
name: Sylvester's determinant theorem
aliases: [weinstein-aronszajn-identity]
msc: [15A15]
uses: [0C3CFHDN6R, 06DW9BAWB1]
sources:
  - url: https://en.wikipedia.org/wiki/Weinstein–Aronszajn_identity
    locator: lead section
same_as:
  wikidata: Q7660749
  wikipedia: Weinstein–Aronszajn identity
---

## Statement

Let $K$ be a commutative ring, let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix over $K$.
Then
$$\det(I_m + AB) = \det(I_n + BA),$$
where $I_m$ and $I_n$ are [[identity matrices|06DW9BAWB1]] and $\det$ is the [[determinant|00KDZRW4C0]].

## Remarks

The two sides are determinants of different sizes, $m$ and $n$.
For a column $u$ and a row $v^T$, it gives $\det(I_m + u v^T) = 1 + v^T u$.
The statement is also called the Weinstein–Aronszajn identity.

## Explanations

### L1 human

The products $AB$ and $BA$ can have very different sizes.
Still, adding the identity to them gives the same determinant.
This can turn a large determinant into a small one.