Math Doubts

Zero matrix

A matrix that consists of zeros as elements is called a zero matrix.

Introduction

A zero matrix is one type of matrix. In this matrix, all entries (or elements) are zero. Hence, the matrix is called a zero matrix and it is also called a zero vector.

Zero matrix

In mathematics, zero was introduced to represent nothing symbolically and the meaning of a word “null” is also nothing. Hence, a zero matrix is also called a null matrix.

For example, we have some elements but all entries are zero. The zero entries are arranged in $m$ rows and $n$ columns, and it is denoted by a symbol $O$.

A zero matrix of the order $m \times n$ is written in matrix form mathematically as follows.

$O$ $\,=\,$ ${\begin{bmatrix} 0 & 0 & 0 & \cdots & 0\\ 0 & 0 & 0 & \cdots & 0\\ 0 & 0 & 0 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 0 \end{bmatrix}}_{\displaystyle m \times n} $

In this null matrix, the number of rows and columns can be equal or different. It means, a zero matrix can be a rectangular matrix or a square matrix.

Examples

Look at the following examples to understand the concept of a zero matrix.

$(1).\,\,\,\,$ $O$ $\,=\,$ $\begin{bmatrix} 0 \\ \end{bmatrix} $

The matrix $O$ is a null matrix of the order $1 \times 1$. In this matrix, there is only one zero in one row and one column.

$(2).\,\,\,\,$ $O$ $\,=\,$ $\begin{bmatrix} 0 & 0 \\ \end{bmatrix} $

The matrix $O$ is a zero matrix of the order $1 \times 2$. In this rectangular matrix, there are two zeros in one row and two columns.

$(3).\,\,\,\,$ $O$ $\,=\,$ $\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ \end{bmatrix} $

The matrix $O$ is a null matrix of the order $2 \times 2$. In this square matrix, there are four zeros in two rows and two columns.

$(4).\,\,\,\,$ $O$ $\,=\,$ $\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $

The matrix $O$ is a zero vector of the order $3 \times 4$. In this rectangular matrix, there are twelve zeros in three rows and four columns.

The above four examples cleared you, how the zero matrices are formed in mathematics. You can also understand that a zero matrix can be a rectangular matrix or a square matrix.

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