In mathematics, Dodgson condensation or method of contractants is a method of computing the determinants of square matrices. It is named for its inventor, Charles Lutwidge Dodgson (better known by his pseudonym, as Lewis Carroll, the popular author). The method in the case of an n × n matrix is to construct an (n − 1) × (n − 1) matrix, an (n − 2) × (n − 2), and so on, finishing with a 1 × 1 matrix, which has one entry, the determinant of the original matrix. General method This algorithm can be described in the following four steps: 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 ai,j with . One can do this using any operation that one could normally perform without changing the value of the determinant, such as adding a multiple of one row to another. Create an (n − 1) × (n − 1) matrix B, consisting of the determinants of every 2 × 2 submatrix of A. Explicitly, we write 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 so . Let A = B, and B = C. Repeat step 3 as necessary until the 1 × 1 matrix is found; its only entry is the determinant. Examples Without zeros One wishes to find All of the interior elements are non-zero, so there is no need to re-arrange the matrix. We make a matrix of its 2 × 2 submatrices. We then find another matrix of determinants: We must then divide each element by the corresponding element of our original matrix. The interior of the original matrix is , so after dividing we get . The process must be repeated to arrive at a 1 × 1 matrix. Dividing by the interior of the 3 × 3 matrix, which is just −5, gives and −8 is indeed the determinant of the original matrix. With zeros Simply writing out the matrices: Here we run into trouble. If we continue the process, we will eventually be dividing by 0. We can perform four row exchanges on the initial matrix to preserve the determinant and repeat