The effect of exchanging two rows on the determinant of a matrix

Question

Answer: is -6 because of the following:

Answer 1

Indeed, @D.L. answered you since changing the sign of determinant happens when you swap the rows. Here $R_3\leftrightarrow R_4$ so you got that - for the second matrix. This can be seen as a very basic way: $$\begin{vmatrix} a & b & c\\ d & e & f\\ g &h &i\end{vmatrix}=a\begin{vmatrix} e & f\\ h &i\end{vmatrix}-b\begin{vmatrix} d & f\\ g &i\end{vmatrix}+c\begin{vmatrix} d & e\\ g &h \end{vmatrix}=a(ei-fh)-b(di-gf)+c(dh-ge)$$ $$=-a(fh-ei)+b(gf-di)-c(ge-dh)=-a\begin{vmatrix} h &i\\e & f\end{vmatrix}+b\begin{vmatrix} g &i\\d& f\end{vmatrix}-c\begin{vmatrix} g&h \\d& e\end{vmatrix}$$ $$=-\begin{vmatrix} a & b & c\\ g &h &i\\d & e & f\end{vmatrix}$$