---
tag: 02FDWY55AS
kind: statement
role: proposition
name: Rank of a product of matrices
msc: [15A03]
uses: [0EWQEYRVA8, 076JGP6R3T]
sources:
  - url: https://en.wikipedia.org/wiki/Rank_(linear_algebra)
    locator: Properties
---

## Statement

Let $F$ be a field, let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $F$.
Then
$$\rank(AB) \le \min(\rank A, \rank B),$$
where $\rank$ is the [[rank of a matrix|0ZC9YTXZRY]].

## Remarks

The column space of $AB$ is contained in the column space of $A$, so $\rank(AB) \le \rank A$.
The row space of $AB$ is contained in the row space of $B$, so $\rank(AB) \le \rank B$.
If $A$ is invertible, then $\rank(AB) = \rank B$; if $B$ is invertible, then $\rank(AB) = \rank A$.
A lower bound is Sylvester's rank inequality: $\rank A + \rank B - n \le \rank(AB)$.

## Explanations

### L1 human

Multiplying by a matrix can never create new independent directions.
So the rank of a product is at most the rank of each factor.
Multiplying by an invertible matrix keeps the rank.