(https://i.stack.imgur.com/12HCj.jpg)
Hello! As I know when we multiply a determinant by a constant we must multiply the answer also by that constant but in these two examples in one of them I had to and in the other not. Can someone explain this to me?
$ \begin{vmatrix} -5 &-1 &-1 &-1 \\ -2 & 3 & 0 & 0 \\ -2 & 0 & 4 & 0 \\ -2 & 0 & 0 & 5 \\ \end{vmatrix} \overset{(1)}= \begin{vmatrix} -5 &-1 &-1 &-1 \\ -2 & 3 & 0 & 0 \\ -2 & 0 & 4 & 0 \\ -\frac25 & 0 & 0 & 1 \\ \end{vmatrix} = \begin{vmatrix} -\frac{27}5 &-1 &-1 & 0 \\ -2 & 3 & 0 & 0 \\ -2 & 0 & 4 & 0 \\ -\frac25 & 0 & 0 & 1 \\ \end{vmatrix} = \begin{vmatrix} -\frac{27}5 &-1 &-1 \\ -2 & 3 & 0 \\ -2 & 0 & 4 \\ \end{vmatrix}= \begin{vmatrix} 27 & 5 & 5 \\ -2 & 3 & 0 \\ -2 & 0 & 4 \\ \end{vmatrix} = 324+40+30 = 394 $
(1): Last row multiplied by $\frac15$
$ \begin{vmatrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \\ 3 & 4 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ \end{vmatrix} = \begin{vmatrix} 1 & 2 & 3 & 4 \\ 0 &-1 &-2 &-7 \\ 0 &-2 &-8 &-10\\ 0 &-7 &-10&-13 \\ \end{vmatrix} = \begin{vmatrix} -1 &-2 &-7 \\ -2 &-8 &-10\\ -7 &-10&-13 \\ \end{vmatrix} = \begin{vmatrix} 1 & 2 & 7 \\ -2 &-8 &-10\\ -7 &-10&-13 \\ \end{vmatrix} = -\begin{vmatrix} 1 & 2 & 7 \\ 0 &-4 & 4 \\ 0 & 4 & 36 \\ \end{vmatrix} = -(-144-16) = 160 $