Minor (linear Algebra) - Multilinear Algebra Approach

Multilinear Algebra Approach

A more systematic, algebraic treatment of the minor concept is given in multilinear algebra, using the wedge product: the k-minors of a matrix are the entries in the kth exterior power map.

If the columns of a matrix are wedged together k at a time, the k × k minors appear as the components of the resulting k-vectors. For example, the 2 × 2 minors of the matrix

are −13 (from the first two rows), −7 (from the first and last row), and 5 (from the last two rows). Now consider the wedge product

where the two expressions correspond to the two columns of our matrix. Using the properties of the wedge product, namely that it is bilinear and

and

we can simplify this expression to

where the coefficients agree with the minors computed earlier.