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.