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## Precalculus

### Unit 7: Lesson 5

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, times, 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, times, 3 zero matrix: $\qquad O_{3\times 3}=\left[\begin{array}{rrr}0 & 0&0 \\ 0 & 0&0 \\ 0 & 0&0 \end{array}\right]$
2, times, 4 zero matrix: $\qquad O_{2\times 4}=\left[\begin{array}{rrrr}0 & 0 &0&0 \\ 0 & 0&0&0 \end{array}\right]$
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, times, n zero matrix to any m, times, n matrix A, we get matrix A back. In other words, A, plus, O, equals, A and O, plus, A, equals, 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, plus, O, equals, B given that $B=\left[\begin{array}{rr}{-2} &5 &6 \\ 8& 1&8 \end{array}\right]$?
times

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

The opposite of a matrix A is the matrix minus, 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, times, n matrix to its opposite, we get the m, times, n zero matrix. So if A is any matrix, then A, plus, left parenthesis, minus, A, right parenthesis, equals, O and minus, A, plus, A, equals, O.
It is also true that A, minus, A, equals, O. This is because subtracting a matrix is like adding its opposite.

## Investigation: What happens when we multiply a matrix by the scalar $0$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, times, n matrix by the scalar 0, we get the m, times, n zero matrix.
Mathematically, this means that 0, A, equals, 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. $\\a+0=a$)Adding a zero matrix to any matrix A gives back the matrix A. (eg. A, plus, O, equals, O, plus, A, equals, A)
Adding any number to its opposite will give zero. (eg. a, plus, left parenthesis, minus, a, right parenthesis, equals, 0)Adding any matrix to its opposite will give a zero matrix. (e.g. A, plus, left parenthesis, minus, A, right parenthesis, equals, O)
Any number times zero is zero. (e.g a, dot, 0, equals, 0).Scalar multiplication of a matrix by 0 will give a zero matrix. (eg. 0, A, equals, O)
Understanding these connections can help make matrix calculations involving a zero matrix much easier!

## Want to join the conversation?

• 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.
(6 votes)
• how martrices are useful in real life ?
(3 votes)
• 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?
(9 votes)
• what is a singular matrix?
(4 votes)
• A singular matrix is a square matrix(2x2 or 3x3) whose determinant is zero.
(3 votes)
• Review Question 4 above is blank and there are no instructions.
(4 votes)
• 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.
(6 votes)
• What about the subtraction part of it? Let's say, A - (-A)? Is it 2A or will it be undefined？
(2 votes)
• Since taking A - (-A) is the same as taking A + A, the answer will indeed be 2A.
(3 votes)
• how are zero matrices useful??
they seem pretty useless for any equation...
(2 votes)
• If you know about the identity property of addition and subtraction, a zero matrix is just that. You may not know it, but the identity property is used in algebraic problem solving all the time.
(2 votes)
• if we add two matrices of different dimensions, why is it that their sum is undefined?
(2 votes)
• Matrices are added entry-by-entry. If the entries don't match up, there is no natural way to pair them, even if the matrices have the same number of entries.
(2 votes)
• What happens when you divide matrices by Zero?
The same thing? Undefined or infinity or whatever?
(1 vote)
• You cannot divide matrices. If a matrix has an inverse, then you can multiply both sides of an equation by that inverse. So for example, if A, B, C are matrices, A has an inverse, and AB=AC, then you can multiply by A⁻¹ to get
A⁻¹AB=A⁻¹AC
IB=IC
B=C

But the zero matrix has no inverse, so we can't do this.
(2 votes)
• What is the determinant of zero matrix? I can't find it anywhere, just on one site (czech wiki) but theres no citation. I think there might be problem of using this ''map''
det: A ----> |R on the zero element. I am not sure.
Here's that website https://cs.wikipedia.org/wiki/Nulov%C3%A1_matice .
(1 vote)
• Being that the zero matrix is comprised entirely of zeros, and that the determinant is ad-bc, the determinant of a zero matrix should just be zero.
(2 votes)