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## Precalculus

### Course: Precalculus>Unit 7

Lesson 5: Properties of matrix addition & scalar multiplication

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.
PropertyExample
Commutative property of additionA, plus, B, equals, B, plus, A
Associative property of additionA, plus, left parenthesis, B, plus, C, right parenthesis, equals, left parenthesis, A, plus, B, right parenthesis, plus, C
Additive identity propertyFor any matrix A, there is a unique matrix O such that A, plus, O, equals, A.
Additive inverse propertyFor each A, there is a unique matrix minus, A such that A, plus, left parenthesis, minus, A, right parenthesis, equals, O.
Closure property of additionA, plus, B is a matrix of the same dimensions as A and B.

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}{\left[\begin{array}{rr}{\blueD3} &\blueD7 \\ \blueD2& \blueD4 \end{array}\right]}+\left[\begin{array}{rr}{\greenD5} &\greenD2 \\ \greenD8& \greenD1 \end{array}\right]&={\left[\begin{array}{rr}{\blueD3+\greenD5} &\blueD7+\greenD2 \\\blueD2+\greenD8& \blueD4+\greenD1 \end{array}\right]} \\\\ &=\left[\begin{array}{rr}{8} &9 \\ 10& 5 \end{array}\right]\\ \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}{\left[\begin{array}{rr}\blueD2&\blueD7 &\goldD8 \\ \blueD2& \blueD4&\goldD3 \end{array}\right]}+\left[\begin{array}{rr}{\greenD5} &\greenD2 \\ \greenD8& \greenD1 \end{array}\right]&=\text{undefined} \end{aligned}

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: $A+B=B+A$A, plus, B, equals, B, plus, 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, plus, 5, equals, 5, plus, 3.
The following example illustrates this matrix property.
\begin{aligned}{\left[\begin{array}{rr}{\blueD 3} &\blueD 7 \\ \blueD 2& \blueD 4 \end{array}\right]}+\left[\begin{array}{rr}{\greenD5} &\greenD2 \\ \greenD8& \greenD1 \end{array}\right]&={\left[\begin{array}{rr}{\blueD3+\greenD5} &\blueD7+\greenD2 \\\blueD2+\greenD8& \blueD4+\greenD1 \end{array}\right]} \\\\ &={\left[\begin{array}{rr}{\greenD5+ \blueD3} &\greenD2+\blueD7 \\\ \greenD8+\blueD2& \greenD1+\blueD4 \end{array}\right]}&\small{\gray{\text{(Real # addition is commutative.)}}} \\\\\ &=\left[\begin{array}{rr}{\greenD5} &\greenD2 \\ \greenD8& \greenD1 \end{array}\right]+{\left[\begin{array}{rr}{\blueD 3} &\blueD 7 \\ \blueD 2& \blueD 4 \end{array}\right]} \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: $(A+B)+C=A+(B+C)$left parenthesis, A, plus, B, right parenthesis, plus, C, equals, A, plus, left parenthesis, B, plus, C, right parenthesis

This property states that you can change the grouping in matrix addition and get the same result. For example, you can add matrix A to B first, and then add matrix C, or, you can add matrix B to C, and then add this result to A.
This property parallels the associative property of addition for real numbers. For example, left parenthesis, 2, plus, 3, right parenthesis, plus, 5, equals, 2, plus, left parenthesis, 3, plus, 5, right parenthesis.
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, left parenthesis, start color #01a995, 5, end color #01a995, plus, start color #aa87ff, 3, end color #aa87ff, right parenthesis, plus, start color #e07d10, 1, end color #e07d10, equals, start color #01a995, 5, end color #01a995, plus, left parenthesis, start color #aa87ff, 3, end color #aa87ff, plus, start color #e07d10, 1, end color #e07d10, right parenthesis.
Because of this property, we can write down an expression like A, plus, B, plus, 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, plus, O, equals, 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.
• $\left[\begin{array}{rr}{3} &-1 \\ 7& 9 \end{array}\right]+\left[\begin{array}{rr}{0} &0 \\ 0& 0 \end{array}\right]=\left[\begin{array}{rr}{3} &-1 \\ 7& 9 \end{array}\right]$
• $\left[\begin{array}{rr}{0} &0 &{0} \\ 0& 0&{0} \end{array}\right]+\left[\begin{array}{rr}{2} &8 &3 \\ -1& 5&7 \end{array}\right]=\left[\begin{array}{rr}{2} &8 &3 \\ -1& 5&7 \end{array}\right]$
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, plus, 0, equals, 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: $A+(-A)=O$A, plus, left parenthesis, minus, A, right parenthesis, equals, O

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 $A=\left[\begin{array}{rr}{-2} &8 \\ -3& 1 \end{array}\right]$, then $-A=\left[\begin{array}{rr}{2} &-8 \\ 3& -1 \end{array}\right]$.
If we add A to minus, A we get a zero matrix, which illustrates the additive inverse property.
\begin{aligned}A+(-A)&=\left[\begin{array}{rr}{-2} &8 \\ -3& 1 \end{array}\right]+\left[\begin{array}{rr}{2} &-8 \\ 3& -1 \end{array}\right]\\\\&=\left[\begin{array}{rr}{-2+2} &8+(-8) \\ -3+3& 1+(-1) \end{array}\right]\\\\ &=\left[\begin{array}{rr}{0} &0 \\ 0& 0 \end{array}\right] \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.

For the problems below, let A, B, and C be 2, times, 2 matrices.
1) Which of the following matrix expressions are equivalent to left parenthesis, A, plus, B, right parenthesis, plus, C, question mark

2) Which of the following matrix expressions are equivalent to left parenthesis, A, plus, left parenthesis, minus, A, right parenthesis, right parenthesis, plus, B?
Remember A and B are 2, times, 2 matrices.

## Want to join the conversation?

• 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.
• 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?
• 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.
• Can matrices also follow De morgans law?
• yes, we will study about them in up coming class
(1 vote)
• I need the proofs of all 9 properties of addition and scalar multiplication. Can you please help me proof all of them
(1 vote)
• Matrices are defined as having those properties. There is nothing to prove. Those properties are what we use to prove other things about matrices.
• May somebody help with where can i find the proofs for these properties
(1 vote)
• You can prove them on your own, use matrices with easy to add and subtract numbers and give proof
• 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.
(1 vote)
• 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?
(1 vote)
• 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.