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# Zero Matrix

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Last updated date: 12th Sep 2024
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## Null Matrix Definition

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

1. The null matrix has all the elements equal to zero.

2. It can have any number of rows and columns and the number of rows and columns can be unequal.

3. The addition of a null matrix to any given matrix of the same order does not change the value of the given matrix.

4. The multiplication of a null matrix to any given matrix gives zero as the product.

5. A null matrix is called the additive identity because when added to another matrix, the value of the matrix doesn’t change.

6. The sum of a matrix and its negative gives a null matrix.

7. The multiplication of a matrix with a scalar 0 gives a null 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.

## FAQs on Zero Matrix

1. Can we get a zero matrix on the multiplication of two non–zero matrices?

Yes, we can get a zero matrix by multiplying two non–zero matrices. For example, product of $A = \left[ {\begin{array}{*{20}{I}}0&0\\{12}&4\end{array}} \right]$ and $B = \left[ {\begin{array}{*{20}{I}}0&{ - 2}\\0&6\end{array}} \right]$ gives 0.

2. Is a null matrix a diagonal matrix?

A diagonal matrix has non–zero diagonal elements and the off-diagonal elements are 0 but in a null matrix all the elements are 0. Thus, the diagonal matrix is not a null matrix.

3. Is it necessary for a null matrix to be a square matrix?

No, it is not necessary for a square matrix to be a null matrix. A zero matrix can be of any order also with an unequal number of rows and columns.