---
tag: 0ZTKJJDGR7
kind: object
name: Zero matrix
msc: [15A99]
instance_of: [0VD9ZSWWJG]
sources:
  - url: https://en.wikipedia.org/wiki/Zero_matrix
    locator: lead section
same_as:
  wikidata: Q338028
  wikipedia: Zero matrix
---

## Description

Let $R$ be a ring and let $m, n \ge 0$ be integers.
The $m \times n$ *zero matrix* over $R$ is the [[matrix|0VD9ZSWWJG]] $0_{m \times n}$ whose entries are all $0$.
It is the identity element for [[matrix addition|0Y7H3SAVA1]]: $A + 0 = 0 + A = A$ for each $m \times n$ matrix $A$.
For each matrix $B$ with $n$ rows, the product $0_{m \times n} B$ is a zero matrix.

## Remarks

The $2 \times 2$ zero matrix is $\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$.
The zero matrix is the matrix of the zero linear map in any pair of bases.
A product of two nonzero matrices can be a zero matrix, for example $\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}^2 = 0$.

## Explanations

### L1 human

The zero matrix is the table with $0$ in every position.
It plays the role of the number $0$: adding it to a matrix changes nothing.
There is one zero matrix for each size.