How can I prove that A=0 using elementary operations?

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Homework Help Overview

The discussion revolves around a linear algebra problem involving matrices, specifically focusing on proving that a matrix A equals zero under certain conditions related to determinants and elementary operations.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the implications of the determinant condition det(A+B) = det(B) for all matrices B. There is discussion about using elementary matrices and operations to manipulate A and B. Some participants express uncertainty about how to demonstrate the necessary properties of determinants in this context.

Discussion Status

The conversation includes hints and suggestions regarding the use of elementary matrices and the properties of determinants. Participants are actively engaging with the problem, but there is no explicit consensus on the next steps or a clear path to a solution.

Contextual Notes

There is a hint suggesting that the proof can be approached by considering the effects of elementary operations on the matrix A. The original poster has expressed difficulty in developing the proof, indicating a need for further clarification and guidance.

drosales
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I need help with another homework problem

Let n be a positive integer and An*n a matrix such that det(A+B)=det(B) for all Bn*n. Show that A=0

Hint: prove property continues to hold if A is modified by any finite number of row or column elementary operations

It seems obvious that A=0 but I'm having trouble developing the proof. Any help would be great.
 
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Please post homework in the homework forum. I moved it for you now.

A hint for the proof: can you write a row/column operation as an elementary matrix??
 
Yes and the product of the elementary matrices returns
A=E1*E2*..*En

is this what you are referring to?
 
Yes. Let E be an elementary matrix, can you show that

det(EA+B)=det(B)

??
 
Im not quite sure how to show this
 
Hint: B=EE^{-1}B.

Use that det(XY)=det(X)det(Y).
 

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