Associative Property of Matrix Multiplication
Associative Property is also valid for multiplication of matrices. We know that matrices are rectangular arrays of numbers. When three matrices are multiplied their product remains same irrespective of pair of matrices taken for multiplication.
Let’s say we have three matrices A, B and C then associative property of matrix multiplication is given as (A × B) × C = A × (B × C). Let’s understand it with an example
[Tex] A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}, C = \begin{bmatrix} 9 & 10 \\ 11 & 12 \end{bmatrix} [/Tex]
Let Check Associative for above given three matrices
LHS We have (A × B) × C =
[Tex] (A \cdot B) \cdot C = \left(\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \cdot \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}\right) \cdot \begin{bmatrix} 9 & 10 \\ 11 & 12 \end{bmatrix} [/Tex]
[Tex]\begin{bmatrix} 31 & 36 \\ 33 & 38 \end{bmatrix} \cdot \begin{bmatrix} 9 & 10 \\ 11 & 12 \end{bmatrix} [/Tex]
[Tex]= \begin{bmatrix} 31 \cdot 9 + 36 \cdot 11 & 31 \cdot 10 + 36 \cdot 12 \\ 33 \cdot 9 + 38 \cdot 11 & 33 \cdot 10 + 38 \cdot 12 \end{bmatrix} [/Tex]
[Tex]= \begin{bmatrix} 675 & 742 \\ 715 & 786 \end{bmatrix} [/Tex]
On RHS we have A × (B × C) =
[Tex]A \cdot (B \cdot C) = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \cdot \left(\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \cdot \begin{bmatrix} 9 & 10 \\ 11 & 12 \end{bmatrix}\right) [/Tex]
[Tex]= \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix} \cdot \begin{bmatrix} 111 & 122 \\ 151 & 166 \end{bmatrix} [/Tex]
[Tex]= \begin{bmatrix} 2 \cdot 111 + 3 \cdot 151 & 2 \cdot 122 + 3 \cdot 166 \\ 1 \cdot 111 + 4 \cdot 151 & 1 \cdot 122 + 4 \cdot 166 \end{bmatrix} [/Tex]
[Tex]= \begin{bmatrix} 675 & 742 \\ 715 & 786 \end{bmatrix} [/Tex]
Hence, we see that product of matrices on both LHS and RHS are equal. Hence, we say that the Matrix Multiplication Follows Associative Property.
Learn More : Matrix Multiplication
Associative Property
Associative Property states that when adding or multiplying numbers, the way they are grouped by brackets (parentheses) does not affect the sum or product. It is also known as the Associative Law. This property applies to both multiplication and addition.
Let’s learn about Associative Property in detail, including the Property of Addition and Multiplication, along with some solved examples.
Table of Content
- What is Associative Property?
- Associative Property Formula
- Associative Property of Addition
- Associative Property of Multiplication
- Associative Property of Subtraction
- Associative Property of Division
- Associative Property of Matrix Multiplication
- Associative and Commutative Property
- Associative Property Examples
- Practice Questions on Associative Property