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### Course: Class 12 math (India)>Unit 3

Lesson 4: Properties of matrix addition & scalar multiplication

# Intro to zero matrices

Learn what a zero matrix is and how it relates to matrix addition, subtraction, and scalar multiplication.

#### What you should be familiar with before taking this lesson

A matrix is a rectangular arrangement of numbers into rows and columns.
The dimensions of a matrix give the number of rows and columns of the matrix in that order. Since matrix $A$ has $2$ rows and $3$ columns, it is called a $2×3$ matrix.
If this is new to you, you might want to check out our intro to matrices. You should also make sure you know how to add and subtract matrices and how to multiply a matrix by a scalar.

## Definition of zero matrix

A zero matrix is a matrix in which all of the entries are $0$. Some examples are given below.
$3×3$ zero matrix: $\phantom{\rule{2em}{0ex}}{O}_{3×3}=\left[\begin{array}{rrr}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$
$2×4$ zero matrix: $\phantom{\rule{2em}{0ex}}{O}_{2×4}=\left[\begin{array}{rrrr}0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
A zero matrix is indicated by $O$, and a subscript can be added to indicate the dimensions of the matrix if necessary.
Zero matrices play a similar role in operations with matrices as the number zero plays in operations with real numbers. Let's take a look.

## Investigation: What happens when we add a zero matrix?

Recall that to add two matrices, we simply add the corresponding entries.
Now try the following matrix addition problems. Notice that each problem involves the sum of a matrix and a zero matrix.
1)
$\left[\begin{array}{rr}4& 5\\ 1& 3\end{array}\right]+\left[\begin{array}{rr}0& 0\\ 0& 0\end{array}\right]=$

2)
$\left[\begin{array}{rr}0& 0\\ 0& 0\\ 0& 0\end{array}\right]+\left[\begin{array}{rr}-2& 3\\ 4& 8\\ -1& 7\end{array}\right]=$

### The conclusion

When we add the $m×n$ zero matrix to any $m×n$ matrix $A$, we get matrix $A$ back. In other words, $A+O=A$ and $O+A=A$.
Here the dimensions of the zero matrix are not explicitly given. It is understood that the dimensions of the zero matrix match the dimensions of matrix $A$.

### Reflection question

What are the dimensions of the zero matrix in the equation $B+O=B$ given that $B=\left[\begin{array}{rrr}-2& 5& 6\\ 8& 1& 8\end{array}\right]$?
$×$

## Investigation: What happens when we add opposite matrices?

The opposite of a matrix $A$ is the matrix $-A$, where each element in this matrix is the opposite of the corresponding element in matrix $A$.
For example, if $A=\left[\begin{array}{rr}4& 1\\ -6& 2\end{array}\right]$, then $-A=\left[\begin{array}{rr}-4& -1\\ 6& -2\end{array}\right]$.
Now try the following matrix addition problems. Notice that each problem involves the sum of a matrix and its opposite.
3)
$\left[\begin{array}{rr}4& -3\\ 8& 7\end{array}\right]+\left[\begin{array}{rr}-4& 3\\ -8& -7\end{array}\right]=$

4)
$\left[\begin{array}{rrr}-4& 2& 5\\ 1& 3& -2\end{array}\right]+\left[\begin{array}{rrr}4& -2& -5\\ -1& -3& 2\end{array}\right]=$

### The conclusion

When we add any $m×n$ matrix to its opposite, we get the $m×n$ zero matrix. So if $A$ is any matrix, then $A+\left(-A\right)=O$ and $-A+A=O$.
It is also true that $A-A=O$. This is because subtracting a matrix is like adding its opposite.

## Investigation: What happens when we multiply a matrix by the scalar $0$‍ ?

When we multiply a matrix by a scalar, each entry in the matrix is multiplied by the given scalar.
Now try the following matrix scalar multiplication problems. Notice that each problem involves multiplying a matrix by the scalar $0$.
5)
$0\cdot \left[\begin{array}{rr}5& 4\\ 9& 1\end{array}\right]=$

6)
$0\cdot \left[\begin{array}{rrr}-2& 4& 10\\ 7& -1& 5\\ -3& 4& 2\end{array}\right]=$

### The conclusion

When we multiply any $m×n$ matrix by the scalar $0$, we get the $m×n$ zero matrix.
Mathematically, this means that $0A=O$.

## Summary: Comparing the zero matrix to the real number zero

In the investigations above, we saw that a zero matrix behaves much like the real number zero.
In particular, we can make the following connections:
The number zeroThe zero matrix
Adding zero to any number $a$ gives back that number $a$. (eg. $\begin{array}{rl}& \\ & a+0=a\end{array}$)Adding a zero matrix to any matrix $A$ gives back the matrix $A$. (eg. $A+O=O+A=A$)
Adding any number to its opposite will give zero. (eg. $a+\left(-a\right)=0$)Adding any matrix to its opposite will give a zero matrix. (e.g. $A+\left(-A\right)=O$)
Any number times zero is zero. (e.g $a\cdot 0=0$).Scalar multiplication of a matrix by $0$ will give a zero matrix. (eg. $0A=O$)
Understanding these connections can help make matrix calculations involving a zero matrix much easier!

## Want to join the conversation?

• how are zero matrices useful??
they seem pretty useless for any equation...
• They are as useful as the number 0 for integer or real numbers.
• So is any zero matrix equal to any different zero matrix?
Like, is a O 5x2 equal to a O 4x3?
• No. If two matrices have different dimensions, then they are different objects.
• how martrices are useful in real life ?
• Matrices are the best ways to store data. Many of the video games we play use matrices to store our game stats. They use it to alter the object, in 3D space. They use the 3D matrix and 2D matrix to convert it into the different objects as per requirement. It also has many applications in data encryption(scrambling of data), economics and business, construction, architecture,seismic surveys,animation, physics(even quantum physics) and even in organising complicated group dances!
A thing we must keep in mind is that we only have to study this chapter but the mathematicians who developed these subjects have devoted their lives to it. Why would they devote their lives for something which has no relevance?
• Non-zero matrices of different dimensions are undefined. Eg. 2x3 + 3x2.
Is this also true for zero matrices? It seems to me that the result would be "O" according to the above definition of a zero matrix and the first conclusion. It would therefore be defined.
• I think what Paul is referring to is where Sal explicitly tells us that to calculate such a scenario would be undefined, see the link, beginning at :

And to answer his question regarding the implied dimensions of an O matrix, I think it's as stated above, that we make the zero's 'fit' --> "Here the dimensions of the zero matrix are not explicitly given. It is understood that the dimensions of the zero matrix match the dimensions of matrix A."
• what is a singular matrix?
• A matrix that has a row and column of 1. Essentially a single number [3]
• So zero is like a mathematical black hole in matrices?
• perhaps
• What about the subtraction part of it? Let's say, A - (-A)? Is it 2A or will it be undefined？
• Since taking A - (-A) is the same as taking A + A, the answer will indeed be 2A.
• So wait if the matrices A and B are multiplied to give answer AB and both the matrices are 2 by 2, if it was multiplied like B times A will the answer be as same as matrices AB
(1 vote)
• No, it doesn't work like that. Multiplication is not commutative with matrices, unless you are doing simple scalar multiplication. But if you meant scalar multiplication, you wouldn't call both A and B matrices, and your scalar value would not be given in a 2 x 2 matrix.
Let's say we have a matrix A
┌ ┐
 3 2
 -1 5
└ ┘
And a matrix B
┌ ┐
 -4 8
 0 2
└ ┘
If you multiply A x B to get AB, you will get
┌ ┐
 -12 28
 4 2
└ ┘
However, if you multiply B x A to get BA, you will get
┌ ┐
 -20 32
 -2 10
└ ┘
So, no, A x B does not give the same result as B x A, unless either matrix A is a zero matrix or matrix B is a zero matrix. OR, you could load a scalar value into all 4 elements of one of your matrices, and then you would be doing scalar multiplication.