arXiv:1607.05352math.HO
An Elementary Proof of Dodgson's Condensation Method for Calculating Determinants
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Abstract
In 1866, Charles Ludwidge Dodgson published a paper concerning a method for evaluating determinants called the condensation method. His paper documented a new method to calculate determinants that was based on Jacobi's Theorem. The condensation method is presented and proven here, and is demonstrated by a series of examples. The condensation method can be applied to a number of situations, including calculating eigenvalues, solving a system of linear equations, and even determining the different energy levels of a molecular system. The method is much more efficient than cofactor expansions, particularly for large matrices; for a 5 x 5 matrix, the condensation method requires about half as many calculations. Zeros appearing in the interior of a matrix can cause problems, but a way around the issue can usually be found. Overall, Dodgson's condensation method is an interesting and simple way to find determinants. This paper presents an elementary proof of Dodgson's method.
Geometric intuition is going to be difficult, but I bet there's a relatively elegant algebraic argument that uses some index-twiddling symmetries. There's a one-page inductive proof (with four pages of introduction) on arXiv: https://arxiv.org/pdf/1607.05352.pdf
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