Dodgson Condensation - The General Method

The General Method

This algorithm can be described in the following 4 steps:

1. Let A be the given n × n matrix. Arrange A so that no zeros occur in its interior. An explicit definition of interior would be all a_i,j with . We can do this using any operation that we could normally perform without changing the value of the determinant, such as adding a multiple of one row to another.
2. Create an (n − 1) × (n − 1) matrix B, consisting of the determinants of every 2 × 2 submatrix of A. Explicitly, we write
3. Using this (n − 1) × (n − 1) matrix, perform step 2 to obtain an (n − 2) × (n − 2) matrix C. Divide each term in C by the corresponding term in the interior of A.
4. Let A = B, and B = C. Repeat step 3 as necessary until the 1 × 1 matrix is found; its only entry is the determinant.