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

.

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

.

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.