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

[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

$$det(EA+B)=det(B)$$

??

 

[drosales]

Im not quite sure how to show this

 

[micromass]

Hint: $B=EE^{-1}B$.

Use that $det(XY)=det(X)det(Y)$.