An Elementary Proof of Dodgson's Condensation Method for Calculating Determinants

TL;DR

This paper provides an elementary proof of Dodgson's condensation method for calculating determinants, demonstrating its efficiency and applications in linear algebra and physics, especially for large matrices.

Contribution

It offers a clear, elementary proof of Dodgson's condensation method, highlighting its advantages and practical applications in various scientific computations.

Findings

The condensation method reduces calculations compared to cofactor expansion.

It is effective for large matrices, requiring about half the calculations.

Zeros in the matrix interior can pose challenges, but solutions exist.

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.