I can't help with a geometrical intuition, but it's interesting to work through what happens with a matrix of 9 scalars.
[a b c]
[d e f]
[g h i]
The determinant of course has six terms. (aei - ahf - bdi + bgf + cdh - cge)
For condensation purposes, we view our 3x3 matrix as a 2x2 matrix instead:
[A B]
[D E]
where each element A, B, D, and E is really a 2x2 matrix. These matrices overlap; the second column of A is the first column of B, and the second row of B is the first row of E. This overlapping is probably related to the need to remove information during the process.
We then take the determinant AE - DB of our 2x2 matrix. If we look back up at the original 3x3 scalar matrix, the only element common to all four of these matrices-of-convenience is e, and that's what we use to "normalize" the determinant back down. The determinant of the original matrix is (|A||E| - |D||B|)/e.
The product of two 2x2 determinants has four terms, so the sum we compute has eight instead of six, but two of them, which just happen to be the two terms with no factor of e, cancel out.
For condensation purposes, we view our 3x3 matrix as a 2x2 matrix instead:
where each element A, B, D, and E is really a 2x2 matrix. These matrices overlap; the second column of A is the first column of B, and the second row of B is the first row of E. This overlapping is probably related to the need to remove information during the process.We then take the determinant AE - DB of our 2x2 matrix. If we look back up at the original 3x3 scalar matrix, the only element common to all four of these matrices-of-convenience is e, and that's what we use to "normalize" the determinant back down. The determinant of the original matrix is (|A||E| - |D||B|)/e.
The product of two 2x2 determinants has four terms, so the sum we compute has eight instead of six, but two of them, which just happen to be the two terms with no factor of e, cancel out.