Adding rows to calculate the determinant. 
Evaluate the determinants given that $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=-6.$



*

*$\begin{vmatrix} a+d & b+e & c+f \\ -d & -e & -f \\ g & h & i \end{vmatrix}$ 

*$\begin{vmatrix} a & b & c \\ 2d & 2e & 2f \\ g+3a & h+3b & i+3c \end{vmatrix}$



Here is what I have tried:
1.
$\begin{vmatrix} a+d & b+e & c+f \\ -d & -e & -f \\ g & h & i \end{vmatrix}\stackrel{\text{add row 2 to row 1}}=\begin{vmatrix} a & b & c \\ -d & -e & -f \\ g & h & i \end{vmatrix}\stackrel{\text{factor out $-1$}}=-\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=-(-6)=6.$ 
2.
$\begin{vmatrix} a & b & c \\ 2d & 2e & 2f \\ g+3a & h+3b & i+3c \end{vmatrix}\stackrel{\text{row 1 times -3, add to row 3}}{=}\begin{vmatrix} a & b & c \\ 2d & 2e & 2f \\ g & h & i \end{vmatrix}\stackrel{\text{factor out 2}}{=}2\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=2(-6)=-12.$
Did I do these correctly? 
I've tried some cases with numbers where adding a multiple of one row to another and found that it doesn't not change the value of the determinant. But I can't seem to grasp the intuition as to why this is so from numeric calculations. 
Why is this so? 

Geometrically, the fact that you can add multiples of rows to each other while keeping the determinant the same, is a reflection of the fact that the determinant can be seen as the volume of the parallelepiped with the rows or columns as it's vectors.
Adding a row to a row has the effect of simply skewing the parallelepiped. Much like skewing a parallelogram does not change it's area (since neither the height of base changes length) the same holds for a parallelepiped, hence the determinant stays the same.
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