Two applications of matrix volume (Q2251165)

A formula for the volume of an \(m\times n\) matrix was introduced by \textit{A. Ben-Israel} [Linear Algebra Appl. 167, 87--111 (1992; Zbl 0762.15003)]. He also gave a number of applications, which inspired the authors to look further. In this paper they present two new applications. The first is a criterion for the orthogonality of a matrix. For the second, they show that the consistency of a linear system \(Ax=\beta\) is completely determined by the volumes of the two matrices \(A\) and \(\left( \begin{matrix} A & \beta \\ 0 & 1\end{matrix}\right)\). This leads to a criterion for the consistency of a linear system.