I have a linear algebra question that is really rather simple. I must have a mistaken assumption or understanding, I just don't know where.
Now I have been told that matrix similarity is a formalization of the idea that multiple matrices can be representations of the same linear map between vector spaces (i.e., if two matrices are similar, they are the same linear map written w/r/t two different bases). From the definition of matrix similarity, we have that matrices $A$ and $B$ are similar iff there exists an invertible matrix $S$ satisfying
$$ A = S^{-1}BS. $$
Combine this with the fact that every invertible matrix some product of elementary matrices and just like that you have that the set of matrices similar to $A$, for instance, is exactly the set of matrices obtainable by applying elementary row and column operations to $A$.
I have also been told that matrix similarity is an equivalence relation on the space of square matrices that preserves characteristic polynomial, among other things. But here is the rub: I have also been told that elementary row and column operations preserve the determinant of a matrix, but do not in general preserve its eigenvalues. How can this be, given that similarity transformations (which amount to elementary operations) is guaranteed to preserve characteristic polynomial?
Thanks!