Conflicting defintions/explanations

My professor states in one instance that if is invertible, then when , leading to . Yet later on in class, during a proof involving the multiplicity of invertible matrices, he makes the claim that implies that is invertible when , leading to . I understand this theorem fairly easily, but my professor made a big deal about whether the order of the equation is or , yet he clearly contradicts himself if order is so important. A similar is in once instance, leads to , and then later on states that it is that leads to . Again, I understand fairly easily that is a matrix is an inverse of another matrix, then the inverse of that first matrix is the other matrix, but this professor is pedantic.

So I need clarification on the theorem. If then is invertible and , or does imply is invertible and therefore . My professor is a real hardass about following the proofs EXACTLY as they are stated, but like I said, the two claims that he made seem to contradict themselves if order is so important.