Sorry for asking what may be a stupid question, but I'm really struggling conceptually to understand why adding and subtracting rows in a matrix changes the eigenvalues and eigenvectors but not the determinant. I know that scaling and swapping rows changes both, but I can't find anything on adding and subtracting.

The question I was studying was true/false: A row replacement operation on A does not change the eigenvalues.

I looked on numerous sites and they all said false, but none of them had any justification.

  • $\begingroup$ Cause the caracteristic polynom changes and its roots too. $\endgroup$ – hamam_Abdallah Nov 3 '16 at 14:15

Well, one thing that may happen when you sum rows is that, on a triangular matrix with a zero diagonal entry, you may possibly change the number of zero diagonal entries. This makes new eigenvalues appear: consider the matrices $$\begin{pmatrix}0&1\\ 0&0\end{pmatrix}\quad \begin{pmatrix}0&1\\ 0&1\end{pmatrix}$$

It does not change the determinant, though, because the determinant is multilinear and alternating on the rows, so for instance $$\det \begin{pmatrix} \color{blue}{R_1}+\color{red}{\lambda R_2}\\ \color{red}{R_2}\\\vdots\\ R_m\end{pmatrix}=\det\begin{pmatrix} \color{blue}{R_1}\\ R_2\\\vdots\\ R_m\end{pmatrix}+\color{red}{\lambda}\det\begin{pmatrix} \color{red}{R_2}\\ \color{red}{R_2}\\\vdots\\ R_m\end{pmatrix}=\det\begin{pmatrix} R_1\\ R_2\\\vdots\\ R_m\end{pmatrix}+\color{red}0$$


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