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Linear Algebra: Determinant

by drosales
Tags: algebra, determinant, linear
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drosales
#1
Feb21-12, 10:18 AM
P: 9
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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micromass
#2
Feb21-12, 10:36 AM
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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??
drosales
#3
Feb21-12, 11:03 AM
P: 9
Yes and the product of the elementary matrices returns
A=E1*E2*..*En

is this what you are referring to?

micromass
#4
Feb21-12, 11:05 AM
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Linear Algebra: Determinant

Yes. Let E be an elementary matrix, can you show that

[tex]det(EA+B)=det(B)[/tex]

??
drosales
#5
Feb21-12, 11:49 AM
P: 9
Im not quite sure how to show this
micromass
#6
Feb21-12, 12:13 PM
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Hint: [itex]B=EE^{-1}B[/itex].

Use that [itex]det(XY)=det(X)det(Y)[/itex].


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