[math] \newcommand{\mathds}{\mathbb}[/math]

Fill in all the geometric details in the basic theory of the determinant, by using the same type of arguments as those in the proof of

[[math]] \det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc [[/math]]

which was fully proved in the above, namely geometric manipulations, and Thales.