What Is a Zero Matrix Definition Properties and Examples
A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has ‘m’ rows and ‘n’ columns,, then the matrix's order is \[m \times n\]. Such a matrix can mathematically be represented as \[A = {\left[ {{a_{ij}}} \right]_{m \times n}}\], where the numbers written in the matrix, that is, \[{a_{ij}}\] which belongs to ith row and jth column, are known as the elements of the matrix.
There are different types of matrices like symmetric matrix, skew–symmetric matrix, zero matrix, square matrix, identity matrix, etc. Here we will particularly discuss the zero matrix. A matrix with all the elements as zero is known as a zero matrix or a null matrix.
We can also perform algebraic operations on the matrices: addition, subtraction and multiplication. To add, subtract or multiply two matrices, we first need to check if the required condition is satisfied by order of the given matrices. For addition and subtraction, the order of the matrices should be equal, while in multiplication, the number of columns in the first matrix should be equal to the number of rows in the second matrix.
What is a Zero Matrix?
The matrix which has all the elements equal to zero is known as a zero matrix or a null matrix. A null matrix is denoted by \[{0_{m \times n}}\], where ‘m’ is the number of rows and ‘n’ is the number of columns. Some examples of zero matrix are given below.
\[{0_{3 \times 3}} = \left[ {\begin{array}{*{20}{I}}0&0&0\\0&0&0\\0&0&0\end{array}} \right]\] , \[{0_{2 \times 4}} = \left[ {\begin{array}{*{20}{I}}0&0&0&0\\0&0&0&0\end{array}} \right]\].
Properties of Zero Matrix
The null matrix has all the elements equal to zero.
It can have any number of rows and columns and the number of rows and columns can be unequal.
The addition of a null matrix to any given matrix of the same order does not change the value of the given matrix.
The multiplication of a null matrix to any given matrix gives zero as the product.
A null matrix is called the additive identity because when added to another matrix, the value of the matrix doesn’t change.
The sum of a matrix and its negative gives a null matrix.
The multiplication of a matrix with a scalar 0 gives a null matrix.
Addition of Zero Matrix
When we add zero matrices to any arbitrary matrix A of the same order, we get the answer as matrix A. For example,
Given \[A = \left[ {\begin{array}{*{20}{I}}2&{16}&{ - 12}\\{ - 4}&1&{ - 52}\end{array}} \right]\] adding a null matrix of the same order to A,
\[A + B\]
\[ = \left[ {\begin{array}{*{20}{I}}2&{16}&{ - 12}\\{ - 4}&1&{ - 52}\end{array}} \right] + \left[ {\begin{array}{*{20}{I}}0&0&0\\0&0&0\end{array}} \right]\]
\[= \left[ {\begin{array}{*{20}{I}}2&{16}&{ - 12}\\{ - 4}&1&{ - 52}\end{array}} \right]\]
Addition of Two Opposite Matrices
When we add the negative of a given matrix A to it, we get a null matrix as the answer, that is, A + (-A) = 0
Given \[A = \left[ {\begin{array}{*{20}{I}}{17}&{ - 2}\\{12}&4\end{array}} \right]\], then negative of A is \[ - A = \left[ {\begin{array}{*{20}{I}}{ - 17}&2\\{ - 12}&{ - 4}\end{array}} \right]\]
\[A + \left( { - A} \right)\]
\[= \left[ {\begin{array}{*{20}{I}}{17}&{ - 2}\\{12}&4\end{array}} \right] + \left[ {\begin{array}{*{20}{I}}{ - 17}&2\\{ - 12}&{ - 4}\end{array}} \right]\]
\[= \left[ {\begin{array}{*{20}{I}}{17 - 17}&{ - 2 + 2}\\{12 - 12}&{4 - 4}\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{I}}0&0\\0&0\end{array}} \right]\]
Multiplication of Null Matrix
When we multiply a zero matrix to any arbitrary matrix A of the same order, we get the answer as a null matrix. For example,
Given \[A = \left[ {\begin{array}{*{20}{I}}{ - 1}&7\\2&{ - 7}\\5&{ - 9}\end{array}} \right]\] multiplying a null matrix B of the order $2\times 4$.
\[\begin{array}{I}AB = \left[ {\begin{array}{*{20}{I}}{ - 1}&7\\2&{ - 7}\\5&{ - 9}\end{array}} \right]\left[ {\begin{array}{*{20}{I}}0&0&0&0\\0&0&0&0\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{I}}0&0&0&0\\0&0&0&0\\0&0&0&0\end{array}} \right]\end{array}\]
Interesting Facts
The determinant of a null matrix is 0.
The determinant of a matrix is the scalar value calculated using a square matrix.
A null matrix is a singular matrix.
The matrix that has a determinant equal to zero is known as a null matrix.
The zero matrices behave the same as the real number zero.
The rank of a null matrix is zero.
A null matrix has no non–zero rows or columns. Thus it has no independent rows or columns.
Solved Examples
Q1. Calculate the sum of \[A = \left[ {\begin{array}{*{20}{I}}2&6&{ - 1}\\{ - 4}&{18}&{ - 5}\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}{ - 2}&{ - 6}&1\\4&{ - 18}&5\end{array}} \right]\]. Name the type of matrix we get as an answer.
Ans. \[A + B\]\[ = \left[ {\begin{array}{*{20}{I}}2&6&{ - 1}\\{ - 4}&{18}&{ - 5}\end{array}} \right] + \left[ {\begin{array}{*{20}{I}}{ - 2}&{ - 6}&1\\4&{ - 18}&5\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{I}}{2 + \left( { - 2} \right)}&{6 + \left( { - 6} \right)}&{ - 1 + 1}\\{ - 4 + 4}&{18 + \left( { - 18} \right)}&{ - 5 + 5}\end{array}} \right]\]
\[= \left[ {\begin{array}{*{20}{I}}0&0&0\\0&0&0\end{array}} \right]\]
As all the elements of the obtained matrix are 0. Thus, it is a null matrix.
Q2. Given \[A = \left[ {\begin{array}{*{20}{I}}0&0\\{12}&4\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}0&{ - 2}\\0&6\end{array}} \right]\]. What is the product of matrix A and B?
Ans. AB \[= \left[ {\begin{array}{*{20}{I}}0&0\\{12}&4\end{array}} \right]\left[ {\begin{array}{*{20}{I}}0&{ - 2}\\0&6\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{I}}0&0\\0&{ - 24 + 24}\end{array}} \right]\]
\[ = \left[ {\begin{array}{*{20}{I}}0&0\\0&0\end{array}} \right]\]
Q3. What is the sum of \[A = \left[ {\begin{array}{*{20}{I}}2&6&{ - 1}\\{ - 4}&{18}&{ - 5}\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}0&0&0\\0&0&0\end{array}} \right]\]?
Ans. \[A + B \]\[= \left[ {\begin{array}{*{20}{I}}2&6&{ - 1}\\{ - 4}&{18}&{ - 5}\end{array}} \right] + \left[ {\begin{array}{*{20}{I}}0&0&0\\0&0&0\end{array}} \right]\]
\[= \left[ {\begin{array}{*{20}{I}}2&6&{ - 1}\\{ - 4}&{18}&{ - 5}\end{array}} \right]\]
Practice Questions
Q1. Calculate the sum of \[A = \left[ {\begin{array}{*{20}{I}}{21}&{65}&{ - 1}\\{ - 40}&{180}&{ - 51}\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}{ - 21}&{ - 65}&1\\{40}&{ - 180}&{51}\end{array}} \right]\]. Name the type of matrix we get as an answer.
Ans. \[\left[ {\begin{array}{*{20}{I}}0&0&0\\0&0&0\end{array}} \right]\]
Q2. Given \[A = \left[ {\begin{array}{*{20}{I}}0&0\\6&{ - 18}\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}0&6\\0&2\end{array}} \right]\]. What is the product of matrix A and B?
Ans. \[\left[ {\begin{array}{*{20}{I}}0&0\\0&0\end{array}} \right]\]
Q3. What is the sum of \[A = \left[ {\begin{array}{*{20}{I}}2&6&{11}\\{40}&{185}&5\end{array}} \right]\] and \[B = \left[ {\begin{array}{*{20}{I}}0&0&0\\0&0&0\end{array}} \right]\]?
Ans. \[\left[ {\begin{array}{*{20}{I}}2&6&{11}\\{40}&{185}&5\end{array}} \right]\]
Key Features
The matrix which has all the elements equal to zero is known as a zero matrix or a null matrix.
A null matrix is denoted by \[{0_{m \times n}}\], where ‘m’ is the number of rows and ‘n’ is the number of columns. The values of rows and columns need not be equal.
The addition of a null matrix to any given matrix of the same order does not change the value of the given matrix. Thus, it is known as the additive identity.
The addition of the negative matrix to itself gives a null matrix.
Conclusion
The matrix which has all the elements equal to zero is known as a zero matrix or a null matrix. A null matrix is denoted by \[{0_{m \times n}}\], where ‘m’ is the number of rows and ‘n’ is the number of columns. The values of rows and columns need not be equal. The addition of a null matrix to any given matrix of the same order does not change the value of the given matrix and so it is known as the additive identity. The multiplication of a null matrix to any given matrix gives zero as the product. The zero matrices behave the same as the real number zero.
List of Related Articles
FAQs on Zero Matrix in Linear Algebra with Definition and Uses
1. What is a zero matrix?
A zero matrix is a matrix in which every element is equal to 0. It can have any order (m × n), but all its entries must be zero.
- If the order is 2 × 3, then all 6 elements are 0.
- It is also called a null matrix.
- It is denoted by O or 0 (with order specified if needed).
2. What is the order of a zero matrix?
The order of a zero matrix is the number of rows and columns it contains, written as m × n. A zero matrix can be of any size as long as all entries are zero.
- Example: A 3 × 2 zero matrix has 3 rows and 2 columns.
- Total elements = m × n.
- Every element in that matrix equals 0.
3. What is the difference between a zero matrix and a null matrix?
There is no difference between a zero matrix and a null matrix; both terms mean a matrix whose entries are all zero. The two names are used interchangeably in linear algebra.
- Zero matrix = Null matrix.
- All elements are 0.
- Can be of any order.
4. What is the role of a zero matrix in matrix addition?
The zero matrix acts as the additive identity in matrix addition. When a zero matrix is added to any matrix of the same order, the result is the original matrix.
- If A is any matrix, then A + O = A.
- The zero matrix must have the same order as A.
- This property is similar to adding 0 in real numbers.
5. What happens when you multiply a matrix by a zero matrix?
When a matrix is multiplied by a zero matrix (with compatible order), the result is a zero matrix. Matrix multiplication with a zero matrix always produces zeros.
- If A is m × n and O is n × p, then A × O = O (m × p zero matrix).
- Similarly, O × A = O when multiplication is defined.
- All resulting entries become 0.
6. Is a zero matrix always a square matrix?
No, a zero matrix is not always square; it can be rectangular or square. The only condition is that all its elements must be zero.
- Example of square: 2 × 2 zero matrix.
- Example of rectangular: 3 × 1 zero matrix.
- Square zero matrices are common in determinant and inverse problems.
7. What is the determinant of a zero matrix?
The determinant of a square zero matrix is 0. Since all rows (and columns) contain only zeros, the determinant evaluates to zero.
- Example: For a 2 × 2 zero matrix, det(O) = 0.
- A zero determinant means the matrix is singular.
- Only defined for square matrices.
8. Does a zero matrix have an inverse?
A zero matrix does not have an inverse because its determinant is zero. A matrix must have a non-zero determinant to be invertible.
- det(O) = 0.
- Therefore, it is a singular matrix.
- No matrix B exists such that O × B = I.
9. Can you give an example of a zero matrix?
An example of a zero matrix is a matrix where every entry is 0, such as a 2 × 2 zero matrix. For instance:
- O = [[0, 0], [0, 0]]
- All four elements are zero.
- This is a square zero matrix of order 2 × 2.
10. What are the main properties of a zero matrix?
The main properties of a zero matrix describe its behavior in matrix operations like addition and multiplication. Key properties include:
- Additive identity: A + O = A.
- Multiplication property: A × O = O and O × A = O (if defined).
- Determinant (square case): det(O) = 0.
- No inverse: It is singular.
- All elements are exactly 0.
