Seeking of eigenvalues and eigenvectors of a given matrix

[annoymage]

Homework Statement

in seeking of eigenvalues and eigenvectors of a given matrix A, is it permissible first to simplify A by means of some elementary operation? (that is, are the eigenvalues and eigenvector of A invariant with respect to elementary row operation)? (prove it)

Homework Equations

n/a

The Attempt at a Solution

i want to prove it, but before that i want to translated it correctly

F is a field, v is eigenvector, λ is eigenvalue

Given A$$\in$$M_nxn(F)

if B is row equivalent to A, then there exist unique λ$$\in$$F and v such that
Av=λv=Bv

so, is my translation correct?

 

[HallsofIvy]

No, unfortunately eigenvalues are changed by "row operations" and so "simplifying" a matrix that way does not help.

 

[annoymage]

yes i know, i got counter example, but i wan to try to prove it systematically,

if B is row equivalent to A, then there exist unique λF and v such that
Av=λv=Bv,

so maybe i can proof by contradiction or something, but is that statement really same as the question?

 

[annoymage]

or i simply just give the counterexample? and done proof?

 

[ninty]

Unless you are explicitly asked for a proof, a counterexample is enough

 

[annoymage]

unless it says "for all elementary matrices" then counter example is ok

but it said "for some elementary matrices", right?

now I'm still rereading all my lecture notes on logic, help me if you can, with the logic owhoo,

 

[annoymage]

can i do like this

Suppose A=(E_n...E₂E₃)B and there exist unique v and λ such that Av=λv and Bv=λv

then, when λ and v is unique then λv is unique which imply Av=Bv,

when v is unique, Av=Bv => A=B !

contradict the fact that A=(E_n...E₂E₃)B => A$$\neq$$B

Conclusion, If A=(E_n...E₂E₃)B then Av$$\neq$$λv or Bv$$\neq$$λv for all v and λ

i this really correct? i can't tell whether I'm just doing thing to trivial, help T_T