In the Shoelace formula, the formula for the area of a polygon is $ A = \frac{1}{2} \begin{vmatrix} x_1 & x_2 & x_3 & ... & x_n & x_1\\ y_1 & y_2 & y_3 & ... & y_n & y_1\\ \end{vmatrix} $
I haven't seen this formally discussed, but I would like to ask if the obtaining of a value from this 2 by n + 1 matrix where n is the number of coordinates is technically considered a determinant despite not being a square matrix.
I know that it imitates determinants in the sense that even if it is not a square matrix, a value from it is obtained by a method similar to Basketweave rule. Which is why I would also consider the idea that it is really just an imitation and that the term determinant is just used because of the similarity. This is similar to how phasors and vectors have their terminology mixed up because phasors do imitate vectors even though technically they do not have direction and are still considered scalar quantities.
Also, I see in the internet the area of a triangle as the determinant of a square matrix as follows
$ A = \frac{1}{2} \begin{vmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \\ 1 & 1 & 1 \\ \end{vmatrix} $
I would like to ask if for polygons with more sides, the 2 by n + 1 matrix can be converted into an equivalent square matrix similar to this.