(adsbygoogle = window.adsbygoogle || []).push({}); Determinant formula in monomials -- can it be generalized?

I ran across this question in one of the Usenet groups (fr.sci.maths), and after doing a double take and realizing what was actually being asked I realized I don't know the answer, and after searching a bit I haven't turned it up, so I thought I'd post it here.

A cute formula for the determinant of a 2x2 matrix is:

[tex]

\left| \begin{matrix}

a & b \\

c & d

\end{matrix}

\right| ~=~ {1 \over 2} \cdot \left( (a + b) (d - c) + (a - b) (d + c) \right)

[/tex]

Of course this is the "usual" ab-cd formula factored into a sum of products of monomials. The question was whether thisformof the formula can be generalized to higher orders?

I'm well aware of the formula using expansion in cofactors, and I know you can expand the determinant as a sum of products of all permutations of selections of one element from each row (or column), and I know it's the (signed) hypervolume of the hyperrectangle spanned by the column vectors. But I don't know any way in general to expand it as a sum of products of monomials analogous to this formula, and searching Google, the CRC Math Tables, and Artin's "Algebra" didn't turn anything up.

This looks vaguely like the formula for Vandermonde's determinant, but the relationship, if any, is too vague to tell me much.

Any ideas or comments will be appreciated.

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# Determinant formula in monomials - can it be generalized?

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