Associative Property of Matrix

Introduction 
Associative property is property of some binary operation ( operation that involves two operands )

ASSOCIATIVITY means within an expression containing two or more occurences in a row of the same associative operator, the order in which the operation are performed does not matter as long as the same sequence of the operands is not changed.

In associative law if we change or re-arrange the parenthesis the result will not change.

Both commutative law and associative law are different. In commutative law order in which operands appears can be changed which is not allowed in case of associative law.

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Associative Property of Matrix Addition
Associative property of  matrix addition states that

Let  A, B and C be n * n matrices then,

( A + B ) + C = A + ( B + C )

statement :   ( A + B ) + C = A + ( B + C )

let us prove the above statement using example

let  A =           `[[a,b],[c,d]]`                                   

let B =   `[[a,b],[c,d]]`


and  C be

`[[a,b],[c,d]]`

To prove :  ( A + B ) + C = A + ( B + C )

so let us first see  ( A + B ) + C

A + B = `[[a,b],[c,d]]+[[a,b],[c,d]]`

( A + B ) + C = ` [[2a,2b],[2c,2d]]+[[a,b],[c,d]] = [[3a,3b],[3c,3d]]` 


next find  A + ( B + C )

( B + C ) = `[[2a,2b],[2c,2d]]`

A + (B + C) =`[[a,b],[c,d]] + [[2a,2b],[2c,2d]] =[[3a,3b],[3c,3d]]`

solving  ( A + B ) + C =  A  + ( B + C ) we came to know that in associative property of multiplication even if we change or rearrange the parenthesis the value of the matrix does not change. It remains same.

Associative Property of  Multiplication

Let  A , B  , C  be n* n matrices . Then ( A B ) C = A ( B C )

A = `[[a,b],[c,d]]` B = `[[a,b],[c,d]]` C = `[[a,b],[c,d]]`

first let us find  (AB )C

(AB) = `[[a,b],[c,d]]``[[a,b],[c,d]]`

` =[[a2+bc,ab+bd],[ac+bd,bc+d2]]`

(AB )C = `[[a2+bc,ab+bd],[ac+bd,bc+d2]]` * `[[a,b],[c,d]]`

= `[[a3+bc2+a2b+bcd,a2b+bcd+ab2+bd2],[a2c+bcd+abc+d2c,ab+bd2+b2c+d3]]`    

Next fine  A ( B C )

(BC) = `[[a,b],[c,d]] *[[a,b],[c,d]]`

= `[[a2+bc,ab+bd],[ac+bd,bc+d2]]`

A(BC) = `[[a2+bc,ab+bd],[ac+bd,bc+d2]]` * `[[a,b],[c,d]]`

= `[[a3+bc2+a2b+bcd,a2b+bcd+ab2+bd2],[a2c+bcd+abc+d2c,ab+bd2+b2c+d3]]`

Solving  ( A B ) C and  A ( B C ) we can observe that even if we rearrange the parenthesis the value does not change. This shows the associative property of multiplication.
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