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 $\lVert\mathbf{x}\rVert$?
 A: In 1841, Cayley published the first English contribution to the theory of determinants. In this paper he used two vertical lines on either side of the array to denote the determinant, a notation which has now become standard.
Update:
Thanks to @HansLundmark for the reference to Cajori's, A History of Mathematical Notations, section 462. Modern Notations, where the text states: 
"-A notation which has rightly enjoyed great popularity because of its objective presentation of the elements composing a determinant, in convenient arrangement for study, was given in 1841 by Cayley..."...
"The first occurrence of Cayley's vertical-line notation for determinants and double vertical-line notation for matrices in Grelle's
Journall is in his "Memoirs sur les hyperdeterminants" ; in Liouville's
Journal, there appeared in 1845 articles by Cayley in which [ ] and
{ } are used in place of the vertical Iines.! The notation { } was
adopted by O. Terquem" in 1848, and by F. Joaehimsthal' in '1849,
who prefixes "det," thus: "det, { }." E. Catalan! wrote "det.
(A, B, G .... )," where A, B, G, .•.. , are the terms along the
principal diagonal. The only objection to Cayley's notation is its
lack of compactness. For that reason, compressed forms are used frequently when objective presentation of the elements is not essential...."
A: 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.
