Zero Matrix (original) (raw)
Last Updated : 23 Jul, 2025
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.
A = \left[\begin{array}{ccc} 1 & -5 & 3\\ 7 & 8 & 4 \end{array}\right]
Table of Content
- What is a Zero Matrix (Null Matrix)?
- Examples of Zero Matrices
- Properties of a Zero Matrix
- Addition of Zero Matrix
- Solved Examples
- FAQs
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.
Zero Matrix
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."
O m×n = \left[\begin{array}{cccccc} 0 & 0 & . & . & . & 0\\ 0 & 0 & . & . & . & 0\\ . & . & . & & & .\\ . & . & & . & & .\\ 0 & 0 & . & . & . & 0 \end{array}\right]_{m\times n}
Examples of Zero Matrices
Some common examples of zero matrices of the different orders are given below:
- Zero Matrix of order (1 × 1) → **P 1,1 **= [0]
- Zero Matrix of order (1 × 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 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:
\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]
**Article Realated to Zero Matrix:
Solved Examples on Zero Matrix
**Example 1: Give an example of a zero matrix that has three rows and four columns.
**Solution:
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."
O3×4 = \left[\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right]_{3\times4}
**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 = \left[\begin{array}{cc} 0 & 0\\ 2 & 0 \end{array}\right] and B = \left[\begin{array}{cc} 0 & 0\\ 1 & 0 \end{array}\right] be two non-zero matrices.
A × B = \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]
A × B = \left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right] = 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."
O2×2 = \left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right]
We know that,
The determinant of a matrix \left[\begin{array}{cc} a & b\\ c & d \end{array}\right] = ad - bc
So, the determinant of O2×2 = 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 = \left[\begin{array}{ccc} 1 & 5 & 9\\ 2 & 8 & 3 \end{array}\right] is a zero matrix.
**Solution:
To prove that, 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]+\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]
=\left[\begin{array}{ccc} 1+0 & 5+0 & 9+0\\ 2+0 & 8+0 & 3+0 \end{array}\right]
= \left[\begin{array}{ccc} 1 & 5 & 9\\ 2 & 8 & 3 \end{array}\right] = A
Hence proved.