# Associative Property of Matrices

Before adding matrices or performing matrix multiplication, it's important to understand the associative property of matrices. Like the associative properties of addition and multiplication, this property of matrices states when you add or multiply any three matrices, the grouping (or association) of the matrices does not affect the result. Let's take a closer look at this property.

## Associative property of addition of matrices

The Associative Property of Addition for Matrices states:

: Let *A*, *B*, and *C* be
$MN$
matrices. Then,
$\left(A+B\right)+C=A+\left(B+C\right)$
.

Let's look at an example. For the matrices below, find $\left(A+B\right)+C$ and $A+\left(B+C\right)$ .

Find $\left(A+B\right)+C$ :

Find $A+\left(B+C\right)$

## Associative property of multiplication of matrices

The Associative Property of Multiplication of Matrices states:

Let *A*, *B*, and *C* be
$n\times n$
matrices. Then,
$\left(AB\right)C=A\left(BC\right)$.

Let's try an example. For the matrices below, find $\left(AB\right)C$ and $A\left(BC\right)$.

Find $\left(AB\right)C$:

Find $A\left(BC\right)$:

## Practice questions on the associative property of matrices

a. Complete the following statement for the Associative Property of Addition for Matrices: Let $A$ , $B$ , and $C$ be $m\times n$ matrices. Then, $\left(A+B\right)+C=\underline{\phantom{\rule{50pt}{0ex}}}$ .

Answer: $A+\left(B+C\right)$

b. Complete the following statement for the Associative Property of
Multiplication of Matrices: Let *A*, *B*, and
*C* be
$n\times n$
matrices. Then,
$\left(AB\right)C=\underline{\phantom{\rule{50pt}{0ex}}}$
.

Answer: $A\left(BC\right)$

c. For

Find $\left(A+B\right)+C$ and $A+\left(B+C\right)$ .

Find $\left(A+B\right)+C$ :

Answer:

Find $A+\left(B+C\right)$ :

Answer:

d. For A =

B = C =Find $\left(AB\right)C$ and $A\left(BC\right)$ .

Find $\left(AB\right)C$ :

Answer:

Find $\left(BC\right)$ :

Answer:

## Topics related to the Associative Property of Matrices

Adding and Subtracting Matrices

Distributive Property of Matrices

## Flashcards covering the Associative Property of Matrices

## Practice tests covering the Associative Property of Matrices

## Learn more about the associative property of matrices

Getting a handle on the Associative Properties of Matrices for both addition and multiplication is important, but not always easy to do. If your student is having trouble grasping the concept of grouping or wants to make sure they fully understand this property for an assignment or test, tutoring is a great option. Working alongside a qualified instructor gives your student the chance to ask pertinent questions and clarify tough concepts. Learn more about the benefits of tutoring by contacting the Educational Directors at Varsity Tutors today.

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