Linear Algebra: Determinant

drosales

I need help with another homework problem

Let n be a positive integer and A_n*n a matrix such that det(A+B)=det(B) for all B_n*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.

micromass

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

Yes and the product of the elementary matrices returns
A=E₁*E₂*..*E_n

is this what you are referring to?

micromass

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

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

??

drosales

Im not quite sure how to show this

micromass

Hint: [itex]B=EE^{-1}B[/itex].

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