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 $0_{m \times n}$ whose entries are all $0$. It is the identity element for matrix addition: $A + 0 = 0 + A = A$ 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.