In the final question, why is the final answer not valid? Isn't B + O equal to B?

As you can see, there is a line in the question that says "Remember A and B are 2 x 2 matrices." The first, second, and third choices fit this restriction, so they are considered valid answers which yield B+O or B for short. The final answer adds a matrix with a dimension of 3 x 2, which is not the same as B (which is only 2 x 2, as stated earlier). Adding these two would be undefined (as shown in one of the earlier videos.), so the last choice isn't a valid answer. Hope this helps! :)

how can i remember names of this properties?

Copy the table below and give a look everyday. You can try a flashcards system, too.

What is the use of a zero matrix?

The zero matrix is just like the number zero in the real numbers. Just like how the number zero is fundamental number, the zero matrix is an important matrix.

I need the proofs of all 9 properties of addition and scalar multiplication. Can you please help me proof all of them

Matrices are defined as having those properties. There is nothing to prove. Those properties are what we use to prove other things about matrices.

If adding a zero matrix is essentially the same as adding the real number zero, why is it not possible to add a 2 by 3 zero matrix to a 2 by 2 matrix?

Because that doesn't change the fact that matrices are added element-by-element, and so they have to have the same dimensions in order to line up. Even if you're just adding zero.

Why do we say "scalar" multiplication? What other things do we multiply matrices by?

We can multiply matrices together, or multiply matrices by vectors (which are just 1xn matrices) as well.

May somebody help with where can i find the proofs for these properties

You can prove them on your own, use matrices with easy to add and subtract numbers and give proof

What do you mean of (Real # addition is commutative)?

It means that if x and y are real numbers, then x+y=y+x. An operation is commutative if you can swap the order of terms in this way, so addition and multiplication of real numbers are commutative operations, but exponentiation isn't, since 2^5≠5^2. You'll see later that matrix multiplication is not commutative.

Main content

Course: Precalculus > Unit 7

Lesson 5: Properties of matrix addition & scalar multiplication

Properties of matrix addition

Google Classroom

Learn about the properties of matrix addition (like the commutative property) and how they relate to real number addition.

In the table below,

A

B

, and

C

are matrices of equal dimensions.

Property	Example
Commutative property of addition	$A + B = B + A$ ‍
Associative property of addition	$A + (B + C) = (A + B) + C$ ‍
Additive identity property	For any matrix $A$ ‍, there is a unique matrix $O$ ‍ such that $A + O = A$ ‍.
Additive inverse property	For each $A$ ‍, there is a unique matrix $- A$ ‍ such that $A + (- A) = O$ ‍.
Closure property of addition	$A + B$ ‍ is a matrix of the same dimensions as $A$ ‍ and $B$ ‍.

This article explores these matrix addition properties.

Matrices and matrix addition

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.

To add two matrices of the same dimensions, simply add the entries in the corresponding positions.

$\begin{aligned} [\begin{array}{rr} 3 & 7 \\ 2 & 4 \end{array}] + [\begin{array}{rr} 5 & 2 \\ 8 & 1 \end{array}] & = [\begin{array}{rr} 3 + 5 & 7 + 2 \\ 2 + 8 & 4 + 1 \end{array}] \\ = [\begin{array}{rr} 8 & 9 \\ 10 & 5 \end{array}] \end{aligned}$ ‍

If any of this is new to you, you should check out the following articles before you proceed:

Dimensions considerations

Notice that the sum of two

2 \times 2

matrices is another

2 \times 2

matrix. In general, the sum of two

m \times n

matrices is another

m \times n

matrix. This describes the closure property of matrix addition.

If the dimensions of two matrices are not the same, the addition is not defined. This is because if

A

is a

2 \times 3

matrix and

B

is a

2 \times 2

matrix, then some entries in matrix

A

will not have corresponding entries in matrix

B

$\begin{aligned} [\begin{array}{rr} 2 & 7 & 8 \\ 2 & 4 & 3 \end{array}] + [\begin{array}{rr} 5 & 2 \\ 8 & 1 \end{array}] & = undefined \end{aligned}$ ‍

Matrix addition & real number addition

Because matrix addition relies heavily on the addition of real numbers, many of the addition properties that we know to be true with real numbers are also true with matrices.

Let's take a look at each property individually.

Commutative property of addition: $A + B = B + A$ ‍

This property states that you can add two matrices in any order and get the same result.

This parallels the commutative property of addition for real numbers. For example,

3 + 5 = 5 + 3

The following example illustrates this matrix property.

\begin{aligned} [\begin{array}{rr} 3 & 7 \\ 2 & 4 \end{array}] + [\begin{array}{rr} 5 & 2 \\ 8 & 1 \end{array}] & = [\begin{array}{rr} 3 + 5 & 7 + 2 \\ 2 + 8 & 4 + 1 \end{array}] \\ = [\begin{array}{rr} 5 + 3 & 2 + 7 \\ 8 + 2 & 1 + 4 \end{array}] & (Real # addition is commutative.) \\ = [\begin{array}{rr} 5 & 2 \\ 8 & 1 \end{array}] + [\begin{array}{rr} 3 & 7 \\ 2 & 4 \end{array}] \end{aligned}

Notice how the commutative property of addition for matrices holds thanks to the commutative property of addition for real numbers!

Associative property of addition: $(A + B) + C = A + (B + C)$ ‍

This property states that you can change the grouping in matrix addition and get the same result. For example, you can add matrix

A

B

first, and then add matrix

C

, or, you can add matrix

B

C

, and then add this result to

A

This property parallels the associative property of addition for real numbers. For example,

(2 + 3) + 5 = 2 + (3 + 5)

Let's justify this matrix property by looking at an example.

In each column we simplified one side of the identity into a single matrix. The two resulting matrices are equivalent thanks to the real number associative property of addition. For example,

(5 + 3) + 1 = 5 + (3 + 1)

Because of this property, we can write down an expression like

A + B + C

and have this be completely defined. We do not need parentheses indicating which addition to perform first, as it doesn't matter!

Additive identity property: $A + O = A$ ‍

A zero matrix, denoted

O

, is a matrix in which all of the entries are

0

Notice that when a zero matrix is added to any matrix

A

, the result is always

A

$[\begin{array}{rr} 3 & - 1 \\ 7 & 9 \end{array}] + [\begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array}] = [\begin{array}{rr} 3 & - 1 \\ 7 & 9 \end{array}]$ ‍
$[\begin{array}{rr} 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}] + [\begin{array}{rr} 2 & 8 & 3 \\ - 1 & 5 & 7 \end{array}] = [\begin{array}{rr} 2 & 8 & 3 \\ - 1 & 5 & 7 \end{array}]$ ‍

These examples illustrate what is meant by the additive identity property; that the sum of any matrix

A

and the appropriate zero matrix is the matrix

A

A zero matrix can be compared to the number zero in the real number system. For all real numbers

a

, we know that

a + 0 = a

. The number

0

is the additive identity in the real number system just like

O

is the additive identity for matrices.

Additive inverse property: $A + (- A) = O$ ‍

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 = [\begin{array}{rr} - 2 & 8 \\ - 3 & 1 \end{array}]

, then

- A = [\begin{array}{rr} 2 & - 8 \\ 3 & - 1 \end{array}]

If we add

A

- A

we get a zero matrix, which illustrates the additive inverse property.

$\begin{aligned} A + (- A) & = [\begin{array}{rr} - 2 & 8 \\ - 3 & 1 \end{array}] + [\begin{array}{rr} 2 & - 8 \\ 3 & - 1 \end{array}] \\ = [\begin{array}{rr} - 2 + 2 & 8 + (- 8) \\ - 3 + 3 & 1 + (- 1) \end{array}] \\ = [\begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array}] \end{aligned}$ ‍

The sum of a real number and its opposite is always

0

, and so the sum of any matrix and its opposite gives a zero matrix. Because of this, we refer to opposite matrices as additive inverses.

Check your understanding

For the problems below, let

A

B

, and

C

2 \times 2

matrices.

1) Which of the following matrix expressions are equivalent to $(A + B) + C ?$ ‍

Because matrix addition is associative, we know that

(A + B) + C = A + (B + C)

Since the grouping of an addition problem doesn't matter, the above expressions are also equal to

A + B + C

Using the commutative and associative properties of matrix addition, we can also show that

(A + B) + C = (C + A) + B

$\begin{aligned} (A + B) + C & = A + (B + C) & Associative property of matrix addition \\ = A + (C + B) & Commutative property of matrix addition \\ = (A + C) + B & Associative property of matrix addition \\ = (C + A) + B & Commutative property of matrix addition \end{aligned}$ ‍

(A + C) + (B + C)

is not equivalent to the others, since there is an extra

C

added to the expression.

The following expressions are equivalent to

(A + B) + C

$A + (B + C)$ ‍
$(C + A) + B$ ‍
$A + B + C$ ‍

2) Which of the following matrix expressions are equivalent to $(A + (- A)) + B$ ‍?
Remember $A$ ‍ and $B$ ‍ are $2 \times 2$ ‍ matrices.

Want to join the conversation?

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jtaylor5c
Posted 8 years ago. Direct link to jtaylor5c's post “In the final question, wh...”
In the final question, why is the final answer not valid? Isn't B + O equal to B?
Button navigates to signup pageButton navigates to signup page
(16 votes)
Answer
- Edward Nacpil
  Posted 8 years ago. Direct link to Edward Nacpil's post “As you can see, there is ...”
  As you can see, there is a line in the question that says "Remember A and B are 2 x 2 matrices."
  
  The first, second, and third choices fit this restriction, so they are considered valid answers which yield B+O or B for short. The final answer adds a matrix with a dimension of 3 x 2, which is not the same as B (which is only 2 x 2, as stated earlier). Adding these two would be undefined (as shown in one of the earlier videos.), so the last choice isn't a valid answer.
  
  Hope this helps! :)
  Comment on Edward Nacpil's post “As you can see, there is ...”
  (84 votes)
Sergey Bobrov
Posted 8 years ago. Direct link to Sergey Bobrov's post “how can i remember names ...”
how can i remember names of this properties?
Button navigates to signup pageComment on Sergey Bobrov's post “how can i remember names ...”
(8 votes)
Answer
- Gustavo Delazeri
  Posted 7 years ago. Direct link to Gustavo Delazeri's post “Copy the table below and ...”
  Copy the table below and give a look everyday. You can try a flashcards system, too.
  Comment on Gustavo Delazeri's post “Copy the table below and ...”
  (13 votes)
john
Posted 7 years ago. Direct link to john's post “The article says, "Becaus...”
The article says, "Because matrix addition relies heavily on the addition of real numbers, many of the addition properties that we know to be true with real numbers are also true with matrices. "

This implies that some of the addition properties of real numbers can't be applied to matrix addition. Anyone know what they are?
Button navigates to signup pageComment on john's post “The article says, "Becaus...”
(6 votes)
Answer
- Patrick
  Posted 7 years ago. Direct link to Patrick's post “For one there is commutat...”
  For one there is commutative multiplication. When you multiply two matrices together in a certain order, you'll get one matrix for an answer. But if you switch the matrices, your product will be completely different than the first one.
  Comment on Patrick's post “For one there is commutat...”
  (8 votes)
John He
Posted 7 years ago. Direct link to John He's post “What is the use of a zero...”
What is the use of a zero matrix?
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(2 votes)
Answer
- Elite Dragon
  Posted 7 years ago. Direct link to Elite Dragon's post “The zero matrix is just l...”
  The zero matrix is just like the number zero in the real numbers. Just like how the number zero is fundamental number, the zero matrix is an important matrix.
  Button navigates to signup page
  (3 votes)
Arnab Chowdhury
Posted 7 years ago. Direct link to Arnab Chowdhury's post “Can matrices also follow ...”
Can matrices also follow De morgans law?
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(3 votes)
Answer
- wwhy5000
  Posted a year ago. Direct link to wwhy5000's post “yes, we will study about ...”
  yes, we will study about them in up coming class
  Button navigates to signup page
  (1 vote)
Anke Schoeman
Posted 7 years ago. Direct link to Anke Schoeman's post “I need the proofs of all ...”
I need the proofs of all 9 properties of addition and scalar multiplication. Can you please help me proof all of them
Button navigates to signup pageComment on Anke Schoeman's post “I need the proofs of all ...”
(1 vote)
Answer
- kubleeka
  Posted 7 years ago. Direct link to kubleeka's post “Matrices are defined as h...”
  Matrices are defined as having those properties. There is nothing to prove. Those properties are what we use to prove other things about matrices.
  Button navigates to signup page
  (4 votes)
Redapple8787
Posted 6 years ago. Direct link to Redapple8787's post “If adding a zero matrix i...”
If adding a zero matrix is essentially the same as adding the real number zero, why is it not possible to add a 2 by 3 zero matrix to a 2 by 2 matrix?
Button navigates to signup pageButton navigates to signup page
(1 vote)
Answer
- kubleeka
  Posted 6 years ago. Direct link to kubleeka's post “Because that doesn't chan...”
  Because that doesn't change the fact that matrices are added element-by-element, and so they have to have the same dimensions in order to line up. Even if you're just adding zero.
  Comment on kubleeka's post “Because that doesn't chan...”
  (4 votes)
Ants Work Diligently
Posted 3 years ago. Direct link to Ants Work Diligently's post “Why do we say "scalar" mu...”
Why do we say "scalar" multiplication? What other things do we multiply matrices by?
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(1 vote)
Answer
- kubleeka
  Posted 3 years ago. Direct link to kubleeka's post “We can multiply matrices ...”
  We can multiply matrices together, or multiply matrices by vectors (which are just 1xn matrices) as well.
  Button navigates to signup page
  (3 votes)
Phumlani Carlit0s
Posted 8 years ago. Direct link to Phumlani Carlit0s's post “May somebody help with wh...”
May somebody help with where can i find the proofs for these properties
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(1 vote)
Answer
- Bleron Qorri
  Posted 8 years ago. Direct link to Bleron Qorri's post “You can prove them on you...”
  You can prove them on your own, use matrices with easy to add and subtract numbers and give proof
  Button navigates to signup page
  (2 votes)
Faria Ashrafi
Posted a year ago. Direct link to Faria Ashrafi's post “What do you mean of (Real...”
What do you mean of (Real # addition is commutative)?
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(1 vote)
Answer
- kubleeka
  Posted a year ago. Direct link to kubleeka's post “It means that if x and y ...”
  It means that if x and y are real numbers, then x+y=y+x.
  An operation is commutative if you can swap the order of terms in this way, so addition and multiplication of real numbers are commutative operations, but exponentiation isn't, since 2^5≠5^2.
  
  You'll see later that matrix multiplication is not commutative.
  Button navigates to signup page
  (2 votes)