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111 LearnersLast updated on November 14, 2025
Null Matrix
A matrix is any rectangular arrangement of numbers. Matrices are usually rectangular, but they can also be square. A null matrix has all its elements equal to zero. Hence, it is also known as the zero matrix. The number of rows and columns in a matrix represents the size of the matrix. This article discusses null matrix, its properties, and applications.
What is Null Matrix (Zero Matrix)?
In a null matrix, all the elements are zeros. It is represented by “0”. The null matrix can be a square or a rectangular matrix. For example, \(A = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \) The zero matrix acts as the additive identity for matrices. This means that if we add a zero matrix to any matrix of the same order, it results in the original matrix (A + 0 = A).
What is the Rank of Zero Matrix?
The rank of a zero matrix is the number of linearly independent rows or columns in the matrix. In a zero matrix, none of the rows or columns are linearly independent, so the rank of a zero matrix is always zero. This is because the column space or row space of a zero matrix contains only the zero vector, meaning it has no dimension.
The nullity of a matrix is the number of independent solutions to the equation Ax = 0. It’s equal to the number of columns minus the rank of the matrix.
How to Add Null Matrix?
In this section, we will learn how to add null matrices. When a matrix A of the order m × n is added to a zero matrix of the same order, the sum is always matrix A. It can be represented as A + 0 = 0 + A = A.
For example, adding A and B
\(A = \begin{bmatrix} 2 & 9 \\ 5 & 4 \end{bmatrix} \quad\text{and}\quad B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
\(A + B = \begin{bmatrix} 2 & 9 \\ 5 & 4 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
\(= \begin{bmatrix} 2 + 0 & 9 + 0 \\ 5 + 0 & 4 + 0 \end{bmatrix} \)
\(= \begin{bmatrix} 2 & 9 \\ 5 & 4 \end{bmatrix} \)
How to Multiply Null Matrix ?
To multiply a null matrix, apply the zero property of multiplication: When multiplying a matrix by zero, it results in a zero matrix.
For example
\(M = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \quad \text{and} \quad N = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
\(M \times N = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \times \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
\(M \times N = \begin{bmatrix} 1 \times 0 & 3 \times 0 \\ 2 \times 0 & 4 \times 0 \end{bmatrix} \)
\(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
Properties of Null Matrix
The properties of zero matrix are used in matrix operations such as addition and multiplication. Here are some properties of null matrix.
- A zero matrix can be square or rectangular
- When a matrix A is added to a zero matrix of the same order or vice versa, the sum will always be matrix A. A + 0 = 0 + A = A
- The product of a zero matrix and another matrix (with compatible dimensions) results in a zero matrix. For e.g., A × 0 = 0 × A = 0
- When a matrix A is subtracted from itself, it results in a zero matrix. A - A = 0
Common Mistakes and How to Avoid Them in Null Matrix
Students often get confused with the null matrix and make errors when working with them. In this section, we will discuss some common mistakes and the ways to avoid them in null matrix.
Mistake 1
Confusing null matrix with identity matrix
Confusing null with identity matrix is a common mistake that students make. They may assume that null matrix’s diagonal contains 1s, which is wrong.
To avoid this confusion, always remember that all the elements in a null matrix are 0s. Only an identity matrix’s diagonal elements are 1s.
Mistake 2
Adding or subtracting null matrix with matrix of different order
Students try to add or subtract a null matrix with another matrix of different order. For example, trying to add 00 00 with 12 75 43 is wrong because the orders of the matrices don’t match, making addition impossible.
So students should always remember that they can only add or subtract a zero matrix with another matrix of the same order.
Mistake 3
Assuming null matrix has an inverse
Students assume that a null matrix has an inverse, which is wrong. To avoid this error, always remember that a null matrix is not invertible.
This is because only non-singular matrices can have an inverse, as their determinant is not zero. In a null matrix, all elements are zero, which means its determinant is also zero.
Mistake 4
Thinking null matrix is always square
Thinking that a null matrix must be a square matrix is wrong. Null matrices can be of any order, for example
\(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)
and
\(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)
are null matrices where the number of elements in rows and columns are not equal. In square matrices, the number of elements in rows and columns are equal.
Mistake 5
Thinking that matrix addition is anti-commutative
Incorrectly assuming that matrix addition is anti-commutative, that is A + 0 ≠ 0 + A, but it is wrong. As matrix addition is commutative, that is A + 0 = 0 + B. for example,
\(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 6 & 1 \\ 7 & 2 \end{bmatrix} \) = \(\begin{bmatrix} 6 & 1 \\ 7 & 2 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
Real-World Applications of Null Matrix
Null matrices have many real-life applications in various fields like engineering and computer imagery. Given below are a few examples where null matrices are used:
- In computer graphics, a null matrix can be used to represent a transformation that maps all points to the origin, effectively resetting or clearing an image matrix.
- In control theory, the null matrix is sometimes used to analyze system stability or to represent parts of a system that are disconnected or inactive.
- In electrical engineering, null matrices are used to show open circuits or inactive parts of a circuit, meaning those parts have zero current or voltage.
Solved Examples on Null Matrix
Problem 1
Give an example 4 × 3 zero matrix?
\(\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \)
Explanation
a 4 × 3 matrix has 4 rows and 3 columns, and all the elements in a zero matrix are 0.
Problem 2
Add M = 00 00 and N = 13 46
\(M + N = \begin{bmatrix} 3 & 6 \\ 1 & 4 \end{bmatrix} \)
Explanation
Given, M = 00 00 and N = 13 46
\(M + N = \begin{bmatrix} 0 + 3 & 0 + 6 \\ 0 + 1 & 0 + 4 \end{bmatrix} \) \(= \begin{bmatrix} 3 & 6 \\ 1 & 4 \end{bmatrix} \)
Problem 3
Find the product of A = 14 28 and B = 00 00
\(A \cdot B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
Explanation
Given,
\(A = \begin{bmatrix} 4 & 8 \\ 1 & 2 \end{bmatrix} \qquad B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
\(A \cdot B = \begin{bmatrix} 0 \times 4 & 0 \times 8 \\ 0 \times 1 & 0 \times 2 \end{bmatrix} \)
\(A \cdot B = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \)
Problem 4
Add M = 00 00 and N = 16 25
\(M + N = \begin{bmatrix} 6 & 5 \\ 1 & 2 \end{bmatrix} \)
Explanation
\(M + N = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 6 & 5 \\ 1 & 2 \end{bmatrix} \)
\(= \begin{bmatrix} 0 + 6 & 0 + 5 \\ 0 + 1 & 0 + 2 \end{bmatrix} = \begin{bmatrix} 6 & 5 \\ 1 & 2 \end{bmatrix} \)
Problem 5
Is P = 00 01 a null matrix?
No, P is not a null matrix
Explanation
P is not a null matrix, because it has non-zero elements.
FAQs on Null Matrix
1.What is a null matrix?
2.What are the types of matrices?
3.What is the order of the zero matrix?
4.What is the rank of a zero matrix?
5.Can a zero matrix be a rectangular matrix?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
