Why cant I row reduce and get the same Eigen values?

There's no reason this will work because row operations don't preserve similarity and so might change the eigenvalues. More explicitly, by performing row operations you move from a matrix $A$ to a matrix $PA$ where $P$ is invertible and in general $PA$ is not similar to $A$.

To see this even more explicitly, assume that $A$ is diagonal with entries $(1,0)$ and multiply the first row by $2$. Clearly the eigenvalues of the resulting matrix will be $(2,0)$ while those of the original matrix were $(1,0)$.

The eigenvalues of a matrix are the solutions to the polynomial $$ \det(A-\lambda I)=0 $$ not the solution of $$ \det(SA-\lambda I)=0 $$ where $S$ is the product of a bunch of elementary matrices. When you row reduce, you are multiplying on the left (usually) by an invertible matrix. The polynomials above will not in general have their roots coincide.