linear algebra courses, is that for square matrices we have if and only if anyway, in this case we call the inverse of and similarly would be the inverse of
so for square matrices the following four statements are equivalent, meaning any of them implies the other three:
for non-square matrices things are different. an matrix could have left inverse, i.e. for some matrix but no right inverse, i.e. for any
matrix it is also possible that has a right inverse but not a left inverse (or of course might have neither left nor right inverse). but if has both left and right inverse, then it has to be
a square matrix, i.e. and then the left and right inverse must be equal.