Prove that if A is an invertible matrix and AB = BC then B = C.

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In summary, the conversation discusses the proof of the statement "if A is an invertible matrix and AB = BC then B = C". The conversation also addresses the misconception that the cancellation law does not hold for matrices, explaining that the theorem being proven only applies to invertible matrices. The suggested approach to prove the statement is to use A^-1 on the equality AB = BC.
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ha9981
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Prove that if A is an invertible matrix and AB = BC then B = C. I thought the way to approach it was to use A^-1 on the equality AB=BC but then I got stuck. I can't get it so that B is on one side and C alone is on the other.

My Try:
AB = BC
A^-1 AB = A^-1 BC
IB = A^-1BC


Also why does B = C not contradict the statement that "the cancellation law doesn't hold for matrices"?
 
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  • #2
Surely, that's not what you want to prove? It's not true. Take A= C and then, for any B, AB= BC because both are just AB.

You must have meant "If AB= AC then B= C". Just multiply both sides on the left by A[sup[-1[/sup].

And it does not contradict the statement "the cancellation law doesn't hold for matrices" because the cancellation law says that "If AB= AC for any non-zero A, then B= C". The theorem you are trying to prove requires that A be invertible. There are many non-zero matrices that are not invertible.
 
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1. What does it mean for a matrix to be invertible?

For a matrix to be invertible, it must have a unique solution when solving for the variables in a system of linear equations. This means that the matrix has a non-zero determinant and all of its rows and columns are linearly independent.

2. Can you provide an example to illustrate this proof?

Sure, let's say we have matrices A, B, and C such that A is invertible and AB = BC. Multiplying both sides by A inverse, we get A inverse AB = A inverse BC. By the associative property of matrix multiplication, this can be rewritten as B = C. Therefore, we have proved that if A is invertible and AB = BC, then B = C.

3. Is this proof specific to matrices or does it apply to other mathematical objects?

This proof is specific to matrices, as the concept of invertibility and matrix multiplication do not apply to other mathematical objects. However, similar principles may apply in other mathematical contexts.

4. Why is it important to prove this statement?

Proving this statement is important because it helps us understand the relationships between matrices and how they behave under multiplication. It also allows us to make generalized statements about invertible matrices and their properties.

5. Are there any exceptions to this proof?

Yes, this proof assumes that all matrices involved are of the same dimension. If the dimensions of the matrices are not compatible, then matrix multiplication cannot be performed and this proof does not hold.

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