Proving non-associativity and matrices

[fubbies]

Prove:
There exists A, B, C ϵ M, we have [[A,B],C] ≠ [[A,[B,C]] and as such, the operation
is not associative.

M=(d e) : d,e,f,g ϵ R
(f g)

M is a set.


I honestly donn't know where to start. The notation confuses me.

 

[HallsofIvy]

Prove:
There exists A, B, C ϵ M, we have [[A,B],C] ≠ [[A,[B,C]] and as such, the operation
is not associative.

M=(d e) : d,e,f,g ϵ R
(f g)

M is a set.


I honestly donn't know where to start. The notation confuses me.

Did you post this on a different forum also? I thought I had responded already.

In any case the crucial point is what you do not give- that "[A, B]" is the "commutator", AB- BA. So [[A, B], C]= [A, B]C- C[A, B]= (AB- BA)C- C(AB- BA)= ABC- BAC- CAB+ CBA.
[A, [B, C]] (NOT [[A,[B,C]]) is equal to A[B, C]- [B, C]A= A(BC- CB)- (BC- CB)A= ABC- ACB- ABC+ CBA. Can you show that these are not the same?