Multilinearity in columns
The axioms and properties of determinants established so far are asymmetric in that they are stated in terms of rows, while similar statements hold for columns. Among the most important properties is the fact that the determinant of is a multilinear function of its rows. Here we intend to establish multilinearity in columns.
Exercise 1. The determinant of is a multilinear function of its columns.
Proof. To prove linearity in each column, it suffices to prove additivity and homogeneity. Let be three matrices that have the same columns, except for the th column. The relationship between the th columns is specified by or in terms of elements for all Alternatively, in Leibniz' formula
for all such that we have
If for a given set there exists only one such that Therefore by (2) we can continue (1) as
Homogeneity is proved similarly, using equations or in terms of elements for all