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
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 .
For the condensation, two definitions are needed.
1. For an
to be the
That is, form all possible
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
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 .
When the first reduced matrix has dimension
, the terms of the matrix
are calculated by finding the determinant for every square matrix of dimension
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
in red by choosing
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.