How to Identify and Work with Zero Matrices in Maths
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.
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FAQs on Zero Matrix Explained: Definition, Examples & Uses
1. What is a zero matrix in mathematics?
A zero matrix, also commonly known as a null matrix, is a matrix in which every element is zero. It is represented by the symbol O. A key characteristic is that a zero matrix can be of any dimension (m x n), meaning it doesn't have to be square. For instance, a 2x3 zero matrix and a 3x3 zero matrix are both valid examples.
2. What are the key properties of a zero matrix?
A zero matrix has several crucial properties that are fundamental to matrix algebra. The primary properties include:
Additive Identity: When a zero matrix (O) is added to any matrix (A) of the same order, the original matrix A remains unchanged (A + O = A).
Multiplication Property: The product of any matrix (A) with a compatible zero matrix (O) results in a zero matrix (A × O = O).
Additive Inverse: For any matrix A, there exists a matrix -A such that their sum is the zero matrix (A + (-A) = O).
Determinant: The determinant of any square zero matrix is always 0.
3. Does a zero matrix have to be a square matrix?
No, a zero matrix is not required to be a square matrix. It can have any dimensions, meaning the number of rows (m) can be different from the number of columns (n). For example, a matrix of order 3x2 with all elements as zero is a valid zero matrix, just as a 4x4 matrix with all zero elements is.
4. How is a zero matrix different from a diagonal matrix?
The main difference lies in their element requirements. A zero matrix must have all its elements equal to zero, regardless of their position. In contrast, a diagonal matrix is a square matrix where only the non-diagonal elements must be zero; the elements on the main diagonal can be any value, including non-zero numbers. While a square zero matrix is technically a special case of a diagonal matrix, most diagonal matrices are not zero matrices.
5. Is it possible to get a zero matrix by multiplying two non-zero matrices?
Yes, unlike with real numbers, the product of two non-zero matrices can result in a zero matrix. This is a unique property of matrix multiplication. For example, consider two non-zero matrices A = [[1, 0], [2, 0]] and B = [[0, 0], [3, 4]]. Their product, AB, will be [[0, 0], [0, 0]], which is a 2x2 zero matrix.
6. How does the role of a zero matrix compare to the role of an identity matrix in matrix algebra?
The zero matrix and the identity matrix serve parallel but opposite functions. The zero matrix (O) is the additive identity, meaning when it is added to another matrix A, the result is A (A + O = A). Its role is analogous to the number 0 in standard addition. Conversely, the identity matrix (I) is the multiplicative identity, meaning when it multiplies another matrix A, the result is A (A × I = A). Its role is analogous to the number 1 in standard multiplication.
7. Can a square zero matrix be considered both symmetric and skew-symmetric?
Yes, a square zero matrix is the only type of matrix that is simultaneously symmetric and skew-symmetric. A matrix A is symmetric if its transpose equals itself (A = AT), and it is skew-symmetric if its transpose equals its negative (AT = -A). A zero matrix (O) fulfills both conditions because its transpose (OT) is still a zero matrix, so O = OT, and its negative (-O) is also a zero matrix, so OT = -O.
