Survey of those properties of determinants that are used in the theory of the last multiplier

Abstract

If one sets

$$\begin{gathered} P = \left( {{a_2} - {a_1}} \right)\left( {{a_3} - {a_1}} \right) \cdots \left( {{a_s} - {a_1}} \right) \hfill \\ \left( {{a_3} - {a_2}} \right) \cdots \left( {{a_s} - {a_2}} \right) \cdots \left( {{a_n} - {a_2}} \right) \hfill \\ \cdots \hfill \\ \cdots \hfill \\ \left( {{a_n} - {a_{n - 1}}} \right) \hfill \\ \end{gathered}$$

then the product P so defined has the property that through a permutation of the quantities a₁, a₂, …, a _n , or what is the same, of the indices 1, 2, …, n, it changes only its sign and not its absolute value. Regarding these permutations, only the following will be referred to.