Null matrix (Zero matrix)

Zero matrix

Definition

A matrix whose all elements are zero, is called a null matrix.

The meaning of null is nothing, which means zero because zero is symbolically used to represent null or nothing in mathematics.

Therefore, if any matrix is having zeros as its elements, the matrix is called as a null matrix. Due to the formation of a matrix by zeros, the null matrix is also called as zero matrix.

Representation

A zero matrix is usually expressed in general form and is denoted by a symbol $O$.

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

It is a matrix, which contains $m$ rows and $n$ columns but zeros are the elements in all the rows and columns. Therefore, it is known as a null matrix of order $m \times n$ and can also represented by $O_{m \times n}$ in mathematics.

A zero matrix can be a rectangular matrix or a square matrix. So, the number of rows can be either equal to or less than or greater than the number of columns.

Example

It is a basic example matrix for rows greater than columns.

$$ O = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} $$

In this matrix, the number of rows and columns are $3$ and $1$ respectively. So, the order of the matrix is $3 \times 1$ and represented by $O_{3×1}$. It is a rectangular matrix

It is another basic example matrix for rows equal to columns.

$$ O = \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix} $$

The matrix contains $2$ rows and $2$ columns. So, it is a matrix of order $2 \times 2$ and represented as $O_{2 \times 2}$. It is a best example for square null matrix.

It is another best example matrix for rows less than columns.

$$ O = \begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} $$

In this zero matrix, the number of rows and columns are $2$ and $3$ respectively. Therefore, it is a matrix of order $2 \times 3$ and denoted as $O_{2 \times 3}$ in mathematics simply.

According to three examples, a null matrix can be either rectangular matrix or square matrix.

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