We recently considered the matrix A square which, of course, it's A times A and it's a natural matrix to consider.
Matrices represents linear transformations and when you have the linear transformations such as rotation you may want to apply twice.
Rotate a body,, take a picture, rotate it again, take another picture.
When you do something like that,  when you apply the same transformations twice, you might have a new linear transformations.
and that new combined linear transformations is ofcourse represented by the matrix A square, so this is a very natural matrix indeed
