# Alice and Jacobi in Determinantland

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This Demonstration shows how to calculate the determinant of a generic square matrix by an alternative method to the classic Laplace expansion. The starting point is Jacobi's theorem, which condenses an matrix into an matrix, where .

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The special case is known as Dodgson's method of condensation, discovered by Charles Dodgson (also known as Lewis Carroll, of Alice in Wonderland fame). The dimension of the matrix is successively reduced by one at each step until it reaches a matrix with entry equal to the determinant [1].

For the condensation, two definitions are needed.

1. For an matrix , define to be the matrix with . That is, form all possible blocks in and take their determinants.

2. Define the interior of , , to be the matrix formed by deleting the first and last rows and first and last columns of .

A step in the condensation consists in dividing term-by-term by .

This method breaks down when the interior matrix contains one or more zeros. It may be possible to remove the zeros from the interior through elementary row and column operations. This Demonstration generates random matrices but excludes cases where the interior contains a zero [2].

When the first reduced matrix has dimension with , the terms of the matrix are calculated by finding the determinant for every square matrix of dimension with

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For the next step, each term is divided by the interior matrix of the initial matrix with its dimension lowered to by taking determinants. After that, if , proceed as before, where every step lowers the dimension by one.

Set the size of the square matrix with "square matrix dimension." Set the dimension of the square condensation matrix with "reduced square matrix dimension." These choices determine how many steps are needed to find the determinant; the lower the dimension, the fewer steps are needed. You can highlight an entry of in red by choosing and on the line starting with .

Set "step" to 1 to understand how the first condensation works. Each step shows the matrix of the step before, the interior matrix of two steps before and the result of this iteration. You can highlight an entry by choosing a row and column. The last step shows the determinant.

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Contributed by: D. Meliga and S. Z. Lavagnino (October 2018)
Additional contributions by: F. Mandirola
Open content licensed under CC BY-NC-SA

## Details

Snapshot 1: lowering the dimension by one through the determinant of several square matrices

Snapshot 2: lowering the dimension from a square matrix to a square matrix; in this case only one step is needed to find the determinant

Snapshot 3: lowering the dimension from a square matrix to a square matrix; in this case there are three steps left to find the determinant

References

[1] D. M. Bressoud, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, New York: Cambridge University Press, 1999.

[2] A. Rice and E. Torrence, "Lewis Carroll’s Condensation Method for Evaluating Determinants," Math Horizons, 14(2), 2006 pp. 12–15. (Aug 29, 2018) www.maa.org/sites/default/files/pdf/upload_library/22/Evans/Horizons-Nov06-p12-15.pdf.

## Permanent Citation

D. Meliga and S. Z. Lavagnino

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