# Null Matrix – Definition, Formula, Properties, Examples | How do you Define a Null Matrix?

Null Matrix or Zero Matrix is also a matrix that consists all of its elements are zero. The Null Matrix will have different rows and columns. Also, the zero matrices can be a square matrix. We can represent the Zero Matrix by the letter ‘O’. It also acts as an additive identity matrix that results in the same matrix when it is added with the matrix having an order of matrix being m x n.

Let us deeply understand the zero matrices by going through the entire article. All the matrices-related articles are given on 10th Grade Math concepts. Check out all the articles to know the complete concept of matrices.

## What is a Null Matrix with Example? | Zero or Null Matrix Definition

A zero matrix is the matrix where all the zero elements are arranged into rows and columns. A null matrix will only have zeros as elements and no other elements present in the zero matrices. It is denoted by ‘O’. The representation of the null matrix is

### Zero Matrix Examples

The zero matrices examples of the different orders are given below:
$$A =\left[ \begin{matrix} 0 \cr 0 \cr \end{matrix} \right]$$
The order of the above zero matrix is 2 × 1.
$$A =\left[ \begin{matrix} 0&0 \cr \end{matrix} \right]$$
The order of the above zero matrix is 1 × 2.
$$A =\left[ \begin{matrix} 0&0 \cr 0&0 \cr \end{matrix} \right]$$
The order of the above zero matrix is 2 × 2.
$$A =\left[ \begin{matrix} 0&0&0 \cr 0&0&0 \cr 0&0&0 \cr \end{matrix} \right]$$
The order of the above zero matrix is 3 × 3.

### Properties of Null Matrix

Below are some of the important properties of the zero matrix.

• The null matrix is also a square matrix.
• Adding a null matrix to any other matrix does not change the added matrix.
• The null matrix may have an unequal number of rows and columns.
• The null matrix is also a singular matrix or zero matrix.
• The multiplication of a null matrix with any other matrix will give you the output as a null matrix.
• The determinant of a null matrix is equal to zero.

### Operations on Zero Matrix

Addition, subtraction, multiplication, etc operations can possible with the null matrix.

• Addition of the Zero Matrix with a non-zero matrix of order m x n, the sum becomes the non-zero matrix of order m x n.
A + O = O + A = A
• If you multiply a non-zero matrix of order m x n with a null matrix, then the output will be a zero matrix.

### Null Matrix Examples

Practice one by one problem and check out the answers and explanations. Easy methods are given to solve the null matrix problems. Go through all the problems and learn the concept deeply.

Example 1.
Give an example of a null matrix having two rows and three columns.

Solution:
Given that a null matrix has two rows and three columns. The null matrix will have all the elements as zero. Therefore, all the elements of two rows and three columns in a null matrix are zero.
$$A =\left[ \begin{matrix} 0&0&0 \cr 0&0&0 \cr \end{matrix} \right]$$

Therefore, the answer is $$A =\left[ \begin{matrix} 0&0&0 \cr 0&0&0 \cr \end{matrix} \right]$$

Example 2:
Prove that the additive identity of $$A =\left[ \begin{matrix} 2&3&4 \cr 6&7&8 \cr \end{matrix} \right]$$ is a null matrix.

Solution:
Given non zero matrix is $$A =\left[ \begin{matrix} 2&3&4 \cr 6&7&8 \cr \end{matrix} \right]$$
To prove that the additive identity of a given non zero matrix is a null matrix, we need to add the given matrix with the null matrix.
$$A =\left[ \begin{matrix} 2&3&4 \cr 6&7&8 \cr \end{matrix} \right]$$ + $$A =\left[ \begin{matrix} 0&0&0 \cr 0&0&0 \cr \end{matrix} \right]$$ = $$A =\left[ \begin{matrix} 2&3&4 \cr 6&7&8 \cr \end{matrix} \right]$$
Thus the addition of a null matrix to the given matrix also gives the same matrix.

Therefore, the null matrix is the additive identity of the given matrix.

Example 3.
Which of the following matrices is the additive identity?
a. Rectangular Matrix
b. Square Matrix
c. Idempotent Matrix
d. Null Matrix

Solution:
The null matrix follows the additive identity.

Example 4.
Find two nonzero matrices whose product is a zero matrix.

Solution:
Let $$A =\left[ \begin{matrix} 0&0 \cr 0&1 \cr \end{matrix} \right]$$ and $$B =\left[ \begin{matrix} 1&0 \cr 0&1 \cr \end{matrix} \right]$$ be two non-zero matrices.

### FAQs on Zero Matrix

1. What is a Null matrix and give an example?

A Null matrix is a matrix that contains all the elements equal to 0. The symbol used to denote the zero matrix is ‘O’. For example, $$A =\left[ \begin{matrix} 0&0 \cr 0&0 \cr \end{matrix} \right]$$ is a zero matrix of order 2 × 2.

2. What is the order of the null matrix?

The order of the null matrix is m x n, where m is the number of rows and n is the number of columns.

3. Is a zero matrix a diagonal matrix?

A diagonal matrix means non-zero elements on its diagonal whereas the off-diagonals elements are zero. But a zero matrix has all its elements as zero. Therefore, the zero matrix is not a diagonal matrix.

4. What is the result of the sum of a zero matrix with a non-zero matrix?

The sum of a non-zero matrix with a zero matrix will result in the non-zero matrix. A + O = A = O + A

5. Is a zero matrix a square matrix?

Yes, a zero matrix is also a square matrix.

### Summary

Know different matrices and their operations and properties on our website. The complete information on the zero matrices is given in this article. All the properties, problems, definitions, and faqs are included along with different examples.

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