examples of non-commutative operations

A standard example of a non-commutative operation is matrix multiplication. Consider the following two integer matrices:

If we compute $AB$ we get

but if we compute $BA$ we have

Since $AB\ne BA$ we conclude that matrix product is not commutative.

Operations do not necessarily have to operate on numbers. Another classic example is function composition. Let $f$ and $g$ be real functions given by

$$f(x)=x^2,g(x)=2x.$$

We see that

$$(f\circ g)(x)=f(g(x))={(2x)}^2=4x^2,$$

but

$$(g\circ f)(x)=g(f(x))=2(x^2)=2x^2.$$

Since we got different functions, we conclude that function composition is not commutative.