Properties of determinants via scalar multiplication

Question

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.

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

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.

Answer 1

Hint: It works also if $k=0$. Then $B$ has a zero row and its determinant is $0$.

Answer 2

No it doesn't change!

Let $A = \begin{bmatrix}2 & 4 \\ 4 & 4 \end{bmatrix}$

$\det(A) =8 - 16 = -8 $

Multiply $R_2$ by $\frac{1}{4}$ a fraction

$B = \begin{bmatrix}2 & 4 \\ 1 & 1 \end{bmatrix}$

$\det(B) = 2- 4 = -2 = \frac{1}{4}(-8) = \frac{1}{4}\det(A)$

If we multiply $R_2$ by zero,

$C = \begin{bmatrix}2 & 4 \\ 0& 0 \end{bmatrix}$

$\det(C) = 0 = 0\det(A)$