# Zero Matrix

A zero matrix, or null matrix, is a matrix whose all elements are zeros. A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an “m by n” matrix when it has “m” rows and “n” columns and is written as an “m Ã— n” matrix. For example, the matrix given below is a “2 Ã— 3” matrix, i.e., a matrix that has two rows and three columns. We have different types of matrices, such as rectangular matrices, square matrices, triangular matrices, symmetric matrices, etc.

[Tex]A = \left[\begin{array}{ccc} 1 & -5 & 3\\ 7 & 8 & 4 \end{array}\right][/Tex]

## What is a Zero Matrix (Null Matrix)?

A zero matrix, or null matrix, is a matrix whose all elements are zeros. As a null matrix has all zeros as its elements, it is referred to as a zero matrix. A zero matrix can be a square matrix, or it can also have an unequal number of rows and columns. A zero matrix is represented as “O.” If we add a zero matrix to another matrix A of the same order, then the resultant matrix is A. So, a zero matrix is known as the additive identity of that particular matrix. The matrix given below represents a zero matrix of order “m by n.”

[Tex]O mÃ—n = \left[\begin{array}{cccccc} 0 & 0 & . & . & . & 0\\ 0 & 0 & . & . & . & 0\\ . & . & . &  &  & .\\ . & . &  & . &  & .\\ 0 & 0 & . & . & . & 0 \end{array}\right]_{m\times n}[/Tex]

## Examples of Zero Matrices

Some common examples of  zero matrices of the different orders are given below:

Zero Matrix of order (1 x 1) â†’ P1,1  = [0]
Zero Matrix of order (1 x 2) â†’ P1,2 = [0, 0]

## Properties of a Zero Matrix

Important properties of a Zero Matrix are:

• A zero matrix can be a square matrix or a rectangular matrix, i.e., it can have an unequal number of rows and columns.
• As the determinant of a zero matrix is zero, a zero matrix is a singular matrix. (Also read about, How to find the Determinant of a Matrix?)
• If a zero matrix is added to another matrix A of the same order, then the resultant matrix is A.

A + O = O + A = A

• If a zero matrix is multiplied by another matrix A, then the resultant matrix is a zero matrix.

A Ã— O = O Ã— A = O

• If any matrix A is subtracted from itself, then the resultant matrix is a zero matrix.

A âˆ’ A = O

• The determinant of a zero matrix or a null matrix is zero.

When a zero matrix of order “m by n” is added to another non-zero matrix A of the same matrix, then the resultant matrix is A.
Let A = [aij]mÃ—n be a non-zero matrix and O be a zero matrix of order “m by n,” then

A + O = O + A = A

Example:

[Tex]\left[\begin{array}{cc} 1 & 2\\ 3 & 4 \end{array}\right]+\left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right]=\left[\begin{array}{cc} 1 & 2\\ 3 & 4 \end{array}\right][/Tex]

## Solved Examples on Zero Matrix

Example 1: Give an example of a zero matrix that has three rows and four columns.

Solution:

The order of a zero matrix that has three rows and four columns is “3 Ã— 4” and all its elements are zero. The matrix given below represents a zero matrix of order “3 Ã— 4.”

O3Ã—4[Tex]\left[\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right]_{3\times4}[/Tex]

Example 2: Prove that if the product of two matrices is a zero matrix, then one of the matrices doesn’t need to be a zero matrix.

Solution:

Let A = [Tex]\left[\begin{array}{cc} 0 & 0\\ 2 & 0 \end{array}\right]  [/Tex] and B = [Tex]\left[\begin{array}{cc} 0 & 0\\ 1 & 0 \end{array}\right]  [/Tex] be two non-zero matrices.

A Ã— B = [Tex]\left[\begin{array}{cc} 0 & 0\\ 2 & 0 \end{array}\right]\left[\begin{array}{cc} 0 & 0\\ 1 & 0 \end{array}\right]=\left[\begin{array}{cc} \left(0\times0+0\times1\right) & \left(0\times0+0\times0\right)\\ \left(2\times0+0\times1\right) & \left(2\times0+0\times0\right) \end{array}\right][/Tex]

A Ã— B = [Tex]\left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right]  [/Tex] = O

Hence proved.

Example 3: Prove that a zero matrix is a singular matrix.

Solution:

To prove that a zero matrix is a singular matrix, let us consider a zero matrix of order “2 Ã— 2.”

O2Ã—2[Tex]\left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right][/Tex]

We know that,

The determinant of a matrix  [Tex]\left[\begin{array}{cc} a & b\\ c & d \end{array}\right]  [/Tex]= ad – bc

So, the determinant of O2Ã—2 = 0 Ã— 0 – 0 Ã— 0 = 0 âˆ’ 0 = 0

We know that a singular matrix is a matrix whose determinant is zero. As the determinant of a zero matrix is zero, a zero matrix is a singular matrix.

Hence proved.

Example 4: Prove that the additive identity of A = [Tex]\left[\begin{array}{ccc} 1 & 5 & 9\\ 2 & 8 & 3 \end{array}\right]  [/Tex] is a zero matrix.

Solution:

To prove that the additive of the given matrix A is a zero matrix, we need to prove that

A + O = A

Given matrix A =

[Tex]\left[\begin{array}{ccc} 1 & 5 & 9\\ 2 & 8 & 3 \end{array}\right]+\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right][/Tex]

=[Tex]\left[\begin{array}{ccc} 1+0 & 5+0 & 9+0\\ 2+0 & 8+0 & 3+0 \end{array}\right][/Tex]

= \left[\begin{array}{ccc} 1 & 5 & 9\\ 2 & 8 & 3 \end{array}\right] = A

Hence proved.

## FAQs on Zero Matrix

Question 1: Define a matrix.

A matrix is defined as a rectangular array of numbers that are arranged in rows and columns.

Question 2: What is a zero matrix?

A zero matrix, or null matrix, is a matrix whose all elements are zeros. As a null matrix has all zeros as its elements, it is referred to as a zero matrix.

Question 3: Is a zero matrix a diagonal matrix?

A diagonal matrix is a square matrix whose non-diagonal elements are zeroes. We know that in a zero matrix all its elements are zeroes. So, we can conclude that a zero matrix is a diagonal matrix.

Question 4: What is the determinant of a zero matrix?