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# 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 $n×n$ 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 $m×n$ 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 $n×n$ 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 $\left(A+B\right)+C$ :

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

d. For A =

B = C =

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

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

Find $\left(BC\right)$ :

## Flashcards covering the Associative Property of Matrices

Precalculus Flashcards