Jumping in without thinking --- it couldn't just be another "extensional identity" in disguise, could it? They can be hard to recognise. See Aitken's little book ("Dets & Mats") WFL On 4/4/07, Richard Schroeppel <rcs@cs.arizona.edu> wrote:

[ [A B] [A B] [A B] ] [ [G H] [I J] [K L] ] [ ] [ [C D] [C D] [C D] ] [ [G H] [I J] [K L] ] = 0. [ ] [ [E F] [E F] [E F] ] [ [G H] [I J] [K L] ]

Each inner 2x2 matrix is replaced with its determinant, and then the determinant of the outer 3x3 is evaluated. It looks more prosaic when you realize that the inner determinants can instead be permanents, or Re((A+Bi)(G+Hi)) or Im, or (A,B)dot(G,H). On the other hand, it looks a little more interesting when you don't apply Det to the inner matrices, but just do Det of the outer 3x3. (The non-commutative multiplication of the inner matrices is fixed by requiring that all the products in the 3x3 det go from left to right, i.e. M1x * M2y * M3z.) Then the det is a 2x2 zero matrix.

Is there a Big Book of these?

Rich rschroe@cs.arizona.edu

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