# Do elementary row operations change eigenvalues?

##### 1 Answer

Yes. For a given matrix

For instance, take the following matrix:

#color(green)(hatA = [(2,2),(0,1)])#

The **eigenvalues** are determined by solving

#\mathbf(hatAvecv = lambdavecv),#

such that

#= |[(lambda,0),(0,lambda)] - [(2,2),(0,1)]|#

#= |(lambda - 2, -2),(0,lambda - 1)|#

#= (lambda-2)(lambda-1) - 0#

From this we acquire the **characteristic equation**:

#=> color(green)((lambda - 2)(lambda - 1) = 0),#

And we get the eigenvalues

# => color(blue)(lambda = 1, 2),#

whose **eigenvectors** are...

#[(lambda - 2, - 2),(0,lambda - 1)][(v_1),(v_2)] = [(0),(0)]#

#= [(-1,-2),(0,0)][(v_1),(v_2)] = [(0),(0)]#

#[(lambda - 2, - 2),(0,lambda - 1)][(v_1),(v_2)] = [(0),(0)]#

#= [(0,-2),(0,1)][(v_1),(v_2)] = [(0),(0)]#

Of course, had you row-reduced

#hatA = [(2,2),(0,1)]#

#stackrel(1/2R_1; -R_2+R_1" ")(->)[(1,0),(0,1)],# where the notation

#cR_i + R_j# implies that#c# times row#i# is added to row#j# and the result is stored into row#j# .

That would give you the characteristic equation

However, without row-reduction, we had gotten two distinct eigenvalues: **not retained** as a result of elementary row operations.