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## Precalculus

### Course: Precalculus>Unit 7

Lesson 11: Properties of matrix multiplication

# Properties of matrix multiplication

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

## Properties of matrix multiplication

In this table, $A$, $B$, and $C$ are $n×n$ matrices, $I$ is the $n×n$ identity matrix, and $O$ is the $n×n$ zero matrix
PropertyExample
The commutative property of multiplication $\text{does not hold!}$$AB\ne BA$
Associative property of multiplication$\left(AB\right)C=A\left(BC\right)$
Distributive properties $A\left(B+C\right)=AB+AC$
$\left(B+C\right)A=BA+CA$
Multiplicative identity property $IA=A$ and $AI=A$
Multiplicative property of zero$OA=O$ and $AO=O$
Dimension propertyThe product of an $m×n$ matrix and an $n×k$ matrix is an $m×k$ matrix.
Let's take a look at matrix multiplication and explore these properties.

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

In matrix multiplication, each entry in the product matrix is the dot product of a row in the first matrix and a column in the second matrix.
If this is new to you, we recommend that you check out our matrix multiplication article.
Here are other relevant articles:

## Matrix multiplication is not commutative

One of the biggest differences between real number multiplication and matrix multiplication is that matrix multiplication is not commutative.
In other words, in matrix multiplication, the order in which two matrices are multiplied matters!

### See for yourselves!

Let's take a look at a concrete example with the following matrices.
$A=\left[\begin{array}{rr}3& 4\\ 1& 2\end{array}\right]$ $\phantom{\rule{1em}{0ex}}$ $B=\left[\begin{array}{rr}6& 2\\ 3& 2\end{array}\right]$
1) Find $AB$ and $BA$.
$AB=$
$BA=$

Notice that the products are not the same! Since $AB\ne BA$, matrix multiplication is not commutative!
Other than this major difference, however, the properties of matrix multiplication are mostly similar to the properties of real number multiplication.

## Associative property of multiplication: $\left(AB\right)C=A\left(BC\right)$‍

This property states that you can change the grouping surrounding matrix multiplication.
For example, you can multiply matrix $A$ by matrix $B$, and then multiply the result by matrix $C$, or you can multiply matrix $B$ by matrix $C$, and then multiply the result by matrix $A$.
When using this property, be sure to pay attention to the order in which the matrices are multiplied, since we know that the commutative property does not hold for matrix multiplication!

## Distributive properties

We can distribute matrices in much the same way we distribute real numbers.
• $A\left(B+C\right)=AB+AC$
• $\left(B+C\right)A=BA+CA$
If a matrix $A$ is distributed from the left side, be sure that each product in the resulting sum has $A$ on the left! Similarly, if a matrix $A$ is distributed from the right side, be sure that each product in the resulting sum has $A$ on the right!

## Multiplicative identity property

The $n×n$ identity matrix, denoted ${I}_{n}$, is a matrix with $n$ rows and $n$ columns. The entries on the diagonal from the upper left to the bottom right are all $1$'s, and all other entries are $0$.
For example:
${I}_{2}=\left[\begin{array}{rr}1& 0\\ 0& 1\end{array}\right]\phantom{\rule{1em}{0ex}}{I}_{3}=\left[\begin{array}{rrr}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\phantom{\rule{1em}{0ex}}{I}_{4}=\left[\begin{array}{rrrr}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
The multiplicative identity property states that the product of any $n×n$ matrix $A$ and ${I}_{n}$ is always $A$, regardless of the order in which the multiplication was performed. In other words, $A\cdot I=I\cdot A=A$.
The role that the $n×n$ identity matrix plays in matrix multiplication is similar to the role that the number $1$ plays in the real number system. If $a$ is a real number, then we know that $a\cdot 1=a$ and $1\cdot a=a$.

## Multiplicative property of zero

A zero matrix is a matrix in which all of the entries are $0$. For example, the $3×3$ zero matrix is ${O}_{3×3}=\left[\begin{array}{rrr}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$.
A zero matrix is indicated by $O$, and a subscript can be added to indicate the dimensions of the matrix if necessary.
The multiplicative property of zero states that the product of any $n×n$ matrix and the $n×n$ zero matrix is the $n×n$ zero matrix. In other words, $A\cdot O=O\cdot A=O$.
The role that the $n×n$ zero matrix plays in matrix multiplication is similar to the role that the number $0$ plays in the real number system. If $a$ is a real number, then we know that $a\cdot 0=0$ and $0\cdot a=0$.

## The dimension property

One property that is unique to matrices is the dimension property. This property has two parts:
1. The product of two matrices will be defined if the number of columns in the first matrix is equal to the number of rows in the second matrix.
2. If the product is defined, the resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
For example, if $A$ is a $3×2$ matrix and if $B$ is a $2×4$ matrix, the dimension property tells us:
• The product $AB$ is defined.
• $AB$ will be a $3×4$ matrix.

Now that you are familiar with matrix multiplication and its properties, let's see if you can use them to determine equivalent matrix expressions.
For the problems below, let $A$, $B$, and $C$ be $2×2$ matrices and let $O$ be the $2×2$ zero matrix.
2) Which of the following expressions are equivalent to $A\left(B+C\right)$?

3) Which of the following expressions are equivalent to ${I}_{2}\left(AB\right)$?

4) Which of the following expressions are equivalent to $O\left(A+B\right)$?

## Want to join the conversation?

• The last exercise (exercise 4), says that 0(A+B) and (A+B)0 give us 0.
But, both final results (the two 0) won't have the same dimensions right?
Because it is not commutative, so: 0(A+B) is not equal to (A+B)0

Am I right? I hope my question was clear
• It was assumed that all A, B and 0 are nxn, therefore 0(A+B)=(A+B)0.
• Hi, everyone.

In the second challenge question, which is given as follows:

2) Which of the following expressions are equivalent to A(B+C)?
Select all that apply.

Selecting i): AB+AC, & ii):A(C+B) will mark the answers right, but selecting " (B+C)A " is wrong? I think it should be right as well, that is, there are overall three answers.

My logic is this that, first we should add 'B' and 'C' and then multiply it by A. We did so in A(C+B), since addition is commutative, ( C+B=B+C ).

So, therefore A(C+B)=(B+C)A, right? I often do this in my Maths book:

(x+a)(x+b)
=x(x+b)+a(x+b)
=x^2+xb+xa+ab
=x^2+(b+a)x+ab

Thanks for reading and you time.
• I still don't get the whole point in making a matrix full of zeros. Isn't it it redundant? Shouldn't the best and easiest way to multiply a matrix to get 0, be to just use the scalar quantity 0 rather than a matrix full of zeros?
• Using the Zero matrix has a lot of use in computing and allows us to compare matrices to algabraic rules
• can a 4x5 multiply a 6x2
• No, and the reason for that is you are going to have an undefined matrix where you can't perform multiplication, such as Matrix 4 x 5 by Matrix 6 x 2 is undefined. So, how can we have a defined matrix? Well, a matrix is defined when the number of columns of the first matrix is equal to the number of rows of the second matrix. For example, let A, B be two matrices. Find out C = AB.

Solution:

A=
| 4 0 9|,

B=
|-1 14 |
| 2 -2 |
|-8 5 |

Now, Matrix A= 1 x 3 and B= 3 x 2, which means that we can go ahead a apply multiplication to AB to get C since Matrix C is defined by having the number of Columns in A equal to the number of Rows in B.
• in the following question which is
Which of the following expressions are equivalent to I2 (AB)
Option
​AB and (AB) I2 were correct i get why AB is correct, however, i m a bit doubtful about the second option for instance if I 2 is a 2 * 2 matrix and A is 2*3 while B is 3*4 well then AB would be 2*4 so I2 ( AB) would be defined but (AB) I2 wouldnt be possible. BTW i dont know what I2 really means but what i have understood after goin through the lecture is that I 2 means identity matrix with 2 rows and 2 columns
• Above all the questions there is a note stating that For the problems below, let A, B, and C be 2×2 matrices and let O be the 2×2 zero matrix
• what is the union and intersection of two matrices?
(1 vote)
• Union and intersection are defined on sets, not on matrices.
• in Q2 of "check your understanding it says:
Which of the following expressions are equivalent to A(B+C)?

Why is (B+C)A wrong?
• Because it is matrix multipliation and you are multiplying rows with columns. Because of that, changing the order changes which numbers get multiplied. Try it out yourself. Take two 2x2 matrices like:
[ 1 2 ]   [ 5 6 ][ 3 4 ]   [ 7 8 ]

And do the dot product, then swap them and do the dot product.
• Yes, it will become BA + CA. Remember, matrix multiplication is not commutative.