Determinant formula in monomials - can it be generalized?

[sal]

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:

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

Of course this is the "usual" ab-cd formula factored into a sum of products of monomials. The question was whether this form of 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.

 

[gel]

you could multiply your matrix on the right by
$$\left( \begin{matrix} 1 & -1 \\ 1 & 1 \end{matrix} \right)$$
(which has determinant 2) then use the usual formula for the determinant to get your expression.
Given any square matrix A, you could write det(A)=det(MAN)/det(MN) for constant matrices M,N to obtain similar expressions.

 

[sal]

Thanks -- I think that leads in the right direction. In particular, I was thinking in terms of factors with just two additive terms, and getting nowhere fast trying to come up with generalizations; what you're suggesting leads to something where, for an NxN matrix, each factor would have N additive terms, which makes a lot more sense!

I'll fiddle with this a bit more in the morning and see if I get any farther with it.

 

[sal]

I'll fiddle with this a bit more in the morning and see if I get any farther with it.

After some fiddling, and playing around with alternative matrices for "N", I posted a response based on Gel's comments on fr.sci.physique (with due credit to Gel). The OP appreciated it. Thread may be found here:

http://groups.google.ca/group/fr.sc..."Expression+du+déterminant"#32069f45d8ab0d28"

It appears that use of "M" in addition to "N", as in "MAN", is unnecessary to narrowly answer the original question; it could, however, be used to produce a result in terms of summations on the components of the column vectors rather than the row vectors.

Thanks again.