1. ## Determinant related proof

Hey here is a question I am currently stuck on:

Let $u = (u_1,u_2,u_3),v = (v_1,v_2,v_3),w = (w_1,w_2,w_3)$ be three vectors in $R^3$. Recall that the orientated volume $V(u,v,w)$ of the parallelpiped spanned by $u,v,w$ equals $(u \times v) \cdot w$.

If $A$ is a $3x3$ matrix prove that

$V(Au,Av,Aw)=det(A)V(u,v,w)$.

The first part of the question (which i got) involved proving that V(u,v,w) = det of the matrix with its first row u, second row v, and third row w. I assume that will likely be useful in this proof.

Honestly not really sure how to do this, other than of course brute forcing and expanding everything and multiplying it together, but i assume there is a much better way to do this. Any help would be great!

2. Hey!

Letting

$B = \left( \begin{array}{ccc} u_{1} & v_{1} & w_{1}\\ u_{2} & v_{2} & w_{2} \\ u_{3} & v_{3} & w_{3} \end{array} \right),$

then I think

$\det{(Au,Av,Aw)} = \det{(AB)}$.

Then, since $\det{(B)} = \det{(B^{t})} = \text{Vol}(u,v,w)$ as you have shown, you just need to prove that $\det{(AB)}=\det{(A)}\det{(B)}.$