Are all algebraic commutative operations always associative? I know that there

Gabriella Sellers

Gabriella Sellers

Answered question

2022-06-25

Are all algebraic commutative operations always associative?
I know that there are many algebraic associative operations which are commutative and which are not commutative.
for example multiplications of matrices as associative operation is not commutative.

Answer & Explanation

Odin Jacobson

Odin Jacobson

Beginner2022-06-26Added 17 answers

Consider the set M n ( R ) and the binary operation A B = 1 2 ( A B + B A )
This isn't associative as
A ( B C ) = A 1 2 ( B C + C B ) = 1 4 ( A B C + A C B + B C A + C B A )
Yet
( A B ) C = 1 4 ( A B C + B A C + C A B + C B A )
and B A C + C A B A C B + B C A for all A , B , C M n ( R )However, A B = 1 2 ( A B + B A ) = 1 2 ( B A + A B ) = B A
So it is commutative.

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