elementary row operation change the eigenvalue of a matrix

Question

I know that elementary row operations may change the eigenvalues of a matrix, but if these operations don't change the determinant, why should change the eigenvalues?? Thanks

Answer 1

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.

Answer 2

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 $\det (B(A-\lambda I)\ne\det (BA-\lambda I)$.