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#1 2007-01-20 08:01:11
- suaviter
- Member
- Registered: 2007-01-20
- Posts: 1
Matrices
Hello! I have a problem proving this property of matrices. I know it's something to do with the associative law of multiplication but I just can't see it yet.
Let A, B and C be n x n matrices.
Prove that if AB = In and CA = In then B = C.
(In being the identity matrix for n x n).
Help!
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#2 2007-01-20 10:31:11
- Ricky
- Moderator
- Registered: 2005-12-04
- Posts: 3,791
Re: Matrices
Depends what class you are taking, but here is the group theory way to do it:
First, note that the set of non-zero matrices, call it M*, forms a group with respect to multiplication.
If A, B, or C is the zero matrix, then it must be that AB = 0 ≠ In or CA = 0 ≠ In. Thus, A, B, and C are all not the zero matrix, and thus, A, B, and C are all in M*.
Since AB = In, then B is the inverse of A. Also, since CA = In, C is the inverse of A. Since inverses in a group must be unique, it must be that B = C.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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