CHAPTER 05.25: SYSTEM OF EQUATIONS: If the inverse of [A] exists, is it unique?
In this segment we will try to prove that if the inverse of A exists that it is unique. Although I posed a question that is it unique and yes it is unique. But we need to now prove it, that the inverse of a A matrix if it exists then it is unique. Letís suppose we have A matrix, of which we, the A matrix is given here. So let B, be inverse of A. And then what weíre going to do is were going to assume C the inverse of A as well. And what that means that we are doing this by contradiction by saying that hey we are saying that B is unique, but if B is not unique then there should be another matrix C, which should also be the inverse of A. And the way to show that whether B is unique or not is to by showing B and C are the same.
Now, letís go and show it, letís go ahead and use this to prove it. We know that B times A is the Identity matrix. And how do we know that? We know that because B is the inverse of A. Thatís the definition itself. And what Iím going to do is Iím going to multiply both of these sides of this by C. So were going to say I times C. Now what is A times C going to give me? Because I just said that hey let C be the inverse of A as well. If C is the inverse of A, then A times C or C times A is also going to be the Identity matrix. So B times I, will be equal to I times C. But we know that if we want to multiply a matrix by a Identity matrix and the multiplication is allowed which is in this case because both of these matrices are M by M and also M by N. So the B matrix multiplied by the Identity matrix will be B itself. And the Identity matrix multiplied by C matrix will be C itself. So this is showing us that hey B and C are the same. So which basically shows that hey that if thereís a matrix A, and it has an inverse, that inverse is unique. And thatís the end of this segment.