Show one way A * B ≠ B * A.?

Question

*cough cough*
Let your wild imagination come to life.

Dang, you're boring people. Nobody said something stupid T-T.

Answer 1

How about when A is a 1x2 matrix and B is a 2x1 matrix?

A*B would result in a 1x1 matrix, but B*A would give you a 2x2 matrix?

Answer 2

Suppose AB is an element of the free group on two generators (which has presentation < A , B | >).

Then AB does not equal BA. If it did, we'd have:

AB=BA

implies

ABA^(-1)B^(-1) = 1

But this is not possible since the presentation contains no relations. Therefore, AB doesn't equal BA.

Answer 3

Vector cross products is another example, in that case,
A × B = -B × A
and if you go in for group theory, you can find tons of non-Abelian groups where A * B ≠ B * A

Answer 4

If either A or B is the square root of any number > 0, it could be the positive root on one side and the negative root on the other side of the inequality.

Answer 5

asking for a counterexample to an algebraic property is silly.

In the set of complex numbers there are no such ways.

So, what are A and B elements of? A group? a ring?
Are they matrices?

Not enough information to answer but as far as common usage goes, impossible.

Answer 6

Multiplication is really about groups.

One group of ten beautiful woman is definitely not equal to ten groups of one beautiful woman.