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. Let’s suppose we
have A matrix, of which we, the A matrix is given here. So let B, be inverse
of A. And then what we’re 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,
let’s go and show it, let’s 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. That’s the definition itself. And what I’m going to do
is I’m 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 there’s a matrix A, and it has an inverse,
that inverse is unique. And that’s the end of this segment. |