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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, times, n matrices, I is the n, times, n identity matrix, and O is the n, times, n zero matrix
PropertyExample
The commutative property of multiplication start color #df0030, start text, d, o, e, s, space, n, o, t, space, h, o, l, d, !, end text, end color #df0030A, B, does not equal, B, A
Associative property of multiplicationleft parenthesis, A, B, right parenthesis, C, equals, A, left parenthesis, B, C, right parenthesis
Distributive properties A, left parenthesis, B, plus, C, right parenthesis, equals, A, B, plus, A, C
left parenthesis, B, plus, C, right parenthesis, A, equals, B, A, plus, C, A
Multiplicative identity property I, A, equals, A and A, I, equals, A
Multiplicative property of zeroO, A, equals, O and A, O, equals, O
Dimension propertyThe product of an m, times, n matrix and an n, times, k matrix is an m, times, 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.
$A=\left[\begin{array}{rr}{3} &4 \\ 1&2 \end{array}\right]$ $\quad$ $B=\left[\begin{array}{rr}{6} &2 \\ 3& 2 \end{array}\right]$
1) Find A, B and B, A.
A, B, equals
B, A, equals

Notice that the products are not the same! Since A, B, does not equal, B, A, 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: $(AB)C=A(BC)$left parenthesis, A, B, right parenthesis, C, equals, A, left parenthesis, B, C, right parenthesis

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 parenthesis, B, plus, C, right parenthesis, equals, A, B, plus, A, C
• left parenthesis, B, plus, C, right parenthesis, A, equals, B, A, plus, C, A
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, times, n identity matrix, denoted I, start subscript, n, end subscript, 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:
$I_2=\left[\begin{array}{rr}{1} &0 \\ 0& 1 \end{array}\right]\quad I_3=\left[\begin{array}{rr}{1} &0 &0 \\ 0& 1&0\\0&0&1 \end{array}\right]\quad I_4=\left[\begin{array}{rr}{1} &0 &0&0 \\ 0& 1&0&0\\0&0&1&0\\0&0&0&1 \end{array}\right]$
The multiplicative identity property states that the product of any n, times, n matrix A and I, start subscript, n, end subscript is always A, regardless of the order in which the multiplication was performed. In other words, A, dot, I, equals, I, dot, A, equals, A.
The role that the n, times, 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, dot, 1, equals, a and 1, dot, a, equals, a.

## Multiplicative property of zero

A zero matrix is a matrix in which all of the entries are 0. For example, the 3, times, 3 zero matrix is $O_{3\times 3}=\left[\begin{array}{rrr}0 & 0&0 \\ 0 & 0&0 \\ 0 & 0&0 \end{array}\right]$.
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, times, n matrix and the n, times, n zero matrix is the n, times, n zero matrix. In other words, A, dot, O, equals, O, dot, A, equals, O.
The role that the n, times, 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, dot, 0, equals, 0 and 0, dot, a, equals, 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 start color #11accd, 3, end color #11accd, times, start color #ed5fa6, 2, end color #ed5fa6 matrix and if B is a start color #ed5fa6, 2, end color #ed5fa6, times, start color #e07d10, 4, end color #e07d10 matrix, the dimension property tells us:
• The product A, B is defined.
• A, B will be a start color #11accd, 3, end color #11accd, times, start color #e07d10, 4, end color #e07d10 matrix.