I'm having trouble connecting all the ideas we're learning in Linear Algebra. On the one hand, I understand how to find determinants, and therefore expansion factors. I also am fairly certain I have a grasp on Cramer's Rule to solve for variables in a system.

However, going through problems in the book, I come across a set of questions that asks me to find areas and volumes using Cramer's rule. Something like this:

Interpret the determinant as an area or volume and as an expansion factor. Use Cramer's Rule.

Find the area of the parallelogram defined by $ \ \left( \begin{array}{cc} 3 \\ 7 \end{array} \right) $ and $ \ \left( \begin{array}{cc} 8 \\ 2 \end{array} \right) $

I was understand the impression that Cramer's rule was only for solving linear systems using determinants. It would be very easy to just answer this with $ \ \left|Det\left( \begin{array}{cc} 3 & 8\\ 7 & 2\end{array} \right)\right| $, but that doesn't feel like I'm using Cramer's rule at all.

I think an answer to this simple question will help me understand how to answer more abstract questions like "Why is the volume of a tetrahedron in R3 defined by v1, v2, and v3 one sixth of the volume of the parallelepiped defined by v1, v2, and v3?". Any guidance would be appreciated.


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