With reference to item (iii), doesn't it have to be an "integer" rather than just a "scalar". Because I have seen instances where the property fails when the multiplication is done by fractions yet I have not seen any text book iterating that. 

[![enter image description here](https://i.stack.imgur.com/eSfmL.png)](https://i.stack.imgur.com/eSfmL.png)

$$
 \begin{vmatrix}
 2 & -3 & 10 \\ 1 & 2 & -2 \\ 0 & 0 & -3 
 \end{vmatrix}
 = -
 \begin{vmatrix}
 1 & 2 & -2 \\ 2 & -3 & 10 \\ 0 & 0 & -3 
 \end{vmatrix}
 = -
 \begin{vmatrix}
 1 & 2 & -2 \\ 0 & -7 & 14 \\ 0 & 0 & -3 
 \end{vmatrix} \\
 =(-1)(-7)(-3)
 =-21
 $$ 

Instead of all this, had I multiplied the second row by $-1/7$ for an instance, the determinant would be different. Can someone explain me what I'm missing here ? Thank you.