TITLE: Proving $(A+B)^2=A^2+2AB+B^2$ wrong using examples
QUESTION [6 upvotes]: Give a counter example to show the following statements are false?
$$(A+B)^2=A^2+2AB+B^2.$$
Would it be possible to use the $2\times 2$ matrices to show the statement the above?

REPLY [17 votes]: Notice that the statement 
$$(A+B)^2=A^2+AB+BA+B^2$$
is true. Therefore the statement you want to disprove is equivalent to 
$$A^2+AB+BA+B^2=A^2+2AB+B^2$$
i.e. 
$$BA=AB$$
It is enough then to find two matrices that don't commute.