My book asks:
Explain in terms of volumes why multiplying all the elements in the first column by constant $k$ multiplies the value of the determinant by $k$ of $(a\space b\space c)$.
To which the answer is:
Stretching factor by $k$ in one direction only multiplies volume by $k$.
So to test it out I carried out:
$\begin{pmatrix} ka_1 & b_1 & c_1 \\ ka_2 & b_2 & c_2 \\ ka_3 & b_3 & c_3 \\ \end{pmatrix} \begin{pmatrix} y \\ x \\ z \\ \end{pmatrix} = \begin{pmatrix} ka_1x+b_1y+c_1z \\ ka_2x+b_2y+c_2z \\ ka_3x+b_3y+c_3z \\ \end{pmatrix}$
Which to me (probably interpretation it wrong) seems to be a small increase in all directions relative to $kx$ which i believe would still increase the determinant by $k$ but doesn't really fit into the answer they've given me, i can't see any broken logic in my reasoning so would appreciate any help understanding my error.
If you have a parallelepiped $\mathcal{P}$, and double just one side - like taking two copies of $\mathcal{P}$ - what happens to the volume?