Why are |vertical lines| used to mark matrix determinants? This notation is sometimes used to denote the determinant:
$$ \begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad-bc$$
Why? Where did this notation come from? Was there any relationship between this notation and the absolute value $|x|$ or the norm $\lVert\mathbf{x}\rVert$?

There is a relationship between the vertical line notation for determinant and the notation $|x|$ for absolute value and $||\mathbf{x}||$ for norm, however I do not know whether this was an intentional decision historically. The absolute value, norm, and determinant, all have at least two things in common.


*

*They are functions mapping a given quantity (a real number, a vector, or a matrix) to a real number.

*They measure the size of something. The absolute value and norm give the distance from the origin to the real number or vector. And the determinant is the factor by which the volume of the unit cube increases under the linear transformation represented by the matrix.
One catch with the analogy is that unlike absolute value and norm, determinants can be negative. In this case however, they are still measuring the factor of volume change. A negative sign simply indicates a change in orientation.
