About a determinant of rectangular \(2\times n\) matrix and its geometric interpretation (Q2568851)

This article is concerned with a geometric interpretation of the determinant of a real \(2\times n\) matrix in terms of the area of an \(n\)-gon in \(\mathbb R^2\) which arises from the columns of the matrix. Also determinants of complex \(2\times n\) matrices are investigated. Here the entries of the first and second row of the matrix are interpreted as vertices of two polygons in the Gaussian plane \(\mathbb C\). There are several papers dealing with a notion of \textit{determinant} of a rectangular matrix: [\textit{C.~E. Cullis} [Matrices and determinoids. Vol. I. Cambridge: University Press, (1913; JFM 44.0171.12)]. \textit{V.~N. Joshi} [Bull. Aust. Math. Soc. 21, 137--146 (1980; Zbl 0421.15007)], \textit{M. Stojaković} [Bull. Soc. R. Sci. Liège 21, 303--305 (1952; Zbl 0048.24901); Bull. Soc. Math. Phys. Serbie 4, Heft 1--2, 9--23 (1952; Zbl 0047.02001); C. R. Acad. Sci., Paris 236, 877--879 (1953; Zbl 0050.01006); C. R. Acad. Sci., Paris 237, 688--690 (1953; Zbl 0051.00801), Bull. Soc. Math. Phys. Serbie 6, 40--55 (1954; Zbl 0057.25102)], \textit{M. Radić} and \textit{R. Sušanj} [Glas. Mat., III. Ser. 27, No.~2, 217--226 (1992; Zbl 0783.15003); Glas. Mat., III. Ser. 29, No.~2, 217--233 (1994; Zbl 0828.15005)] and \textit{M. Radić} [Glas. Mat., III. Ser. 1(21), 17--22 (1966; Zbl 0168.02703)]. Only two of them are cited in the paper under review.