Although this question is already asked and also answered which I am thinking is a bit algebraic..(simply put tough for me ;-) ) hence I had started thinking of in direction of proving it using concept of linearity of Matrix Transformations.
So, before telling my idea, how did I approach I would like to tell that this idea might be a complete disaster so please check my arguments at each step and if it's wrong (totally or partially or any where) plz point it out.
So the idea is,
Let M1 and M2 be 2 Matrices (which can be multiplied, i.e. having no order issues..putting simply so that I can more focus on proof ahead and won't distract the reader from the nitty-gritties)
$$(M1 M2)<=>(M1) (M2)$$
i.e. First applying the transformation M2 and then M1.
Hopefully for obvious reasons this holds from both side of implication.(If this itself is wrong then proof is completely wrong)
Now using this definition,
I am supposed to prove:
$$A(BC)=(AB)C$$
Proof:
Staring from LHS (and hopefully intending to land at RHS)
$(A(BC))$
$<=>(A)(BC)$
$<=>(A)(B)(C)$
$<=>(AB)(C)$
$<=>(AB)C$
Hence, $$A(BC)=(AB)C$$
Is that a right way to prove the associativity ? If yes then what is exactly wrong..? Or How can it be made right?