Proving Two Matrices to be Equal

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Homework Help Overview

The discussion revolves around proving the equality of two matrices, C and D, given certain conditions involving matrix multiplication and identity matrices. The context involves linear algebra concepts related to matrix dimensions and properties.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the implications of the given equations CA=In and AD=Im, questioning the use of inverses due to the non-square nature of the matrices. There is discussion about equating C and D through definitions and properties of matrix multiplication.

Discussion Status

Some participants have suggested considering the matrix CAD and using the associative property as a potential approach. There is a sense of progress as one participant expresses confidence in their understanding of the problem.

Contextual Notes

Participants note that the matrices involved are not square, which raises questions about the applicability of certain matrix properties. The discussion also reflects on the dimensions of the matrices and the implications for proving equality.

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Homework Statement


Suppose that A is an m x n matrix and there exists n x m matrices C and D such that CA=In and AD=Im. Prove that
C=D

Homework Equations

The Attempt at a Solution


Im not sure if I'm on the right path here. However my initial thought is that since the matrices are not square there isn't anything to prove by using inverses. So my guess would be i need to use the definition of matrix multiplication on CA=In and AD=Im and try and equate C and D in some way.
 
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But C and D have the same rows and columns so they could be equal. They don't have to be square.
 
Yes I can see that. But I don't think we can use inverses or inverse properties on this proof because CA and AD are not both the same size matrices.
 
B18 said:
Yes I can see that. But I don't think we can use inverses or inverse properties on this proof because CA and AD are not both the same size matrices.

You don't have to. Think about the matrix CAD. Use the associative property.
 
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Dick said:
You don't have to. Think about the matrix CAD. Use the associative property.
Ok, i think I've got this one nailed down.
CA=In
CAD=InD
C(AD)=D [identity matrix times D is still the matrix D]
since AD=Im
we have C(Im)=D
C=D [identity matrix times C is still the matrix C]
 
B18 said:
Ok, i think I've got this one nailed down.
CA=In
CAD=InD
C(AD)=D [identity matrix times D is still the matrix D]
since AD=Im
we have C(Im)=D
C=D [identity matrix times C is still the matrix C]

Nailed.
 

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