elementary row operation change the eigenvalue of a matrix
Row operations change $A$ into $BA$ for some suitable $B$ with $\det B=1$. It follows that $\det (BA)=\det A$. Regarding eigenvalues, it also follows that $\det (B(A-\lambda I)=\det (A-\lambda I)$, but possibly $\det (B(A-\lambda I)\ne\det (BA-\lambda I)$.
Adding a multiple of one row to another row changes the trace of the matrix (most of the time), but the trace is the sum of the eigenvalues, so (at least one of) the eigenvalues must change, as well.