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 × n” matrix. For example, the matrix given below is a “2 × 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.

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.”

Examples of Zero Matrices

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

Zero Matrix of order (1 x 1) → P_1,1  = [0]
Zero Matrix of order (1 x 2) → P_1,2 = [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 × O = O × A = O

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

A − A = O

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

Addition of Zero Matrix

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]_m×n be a non-zero matrix and O be a zero matrix of order “m by n,” then

A + O = O + A = A

Example:

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 × 4” and all its elements are zero. The matrix given below represents a zero matrix of order “3 × 4.”

O_3×4 = 

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 =  and B =  be two non-zero matrices.

A × B = 

A × B =  = 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 × 2.”

O_2×2 = 

We know that,

The determinant of a matrix  = ad – bc

So, the determinant of O_2×2 = 0 × 0 – 0 × 0 = 0 − 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 =  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 = 

=

= \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.

Answer:

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

Question 2: What is a zero matrix?

Answer:

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?

Answer:

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?

Answer:

As all the elements of a zero matrix are zeroes, the determinant of a zero matrix is zero.