Dodgson condensation

In mathematics, Dodgson condensation is a method of computing the determinants of square matrices. It is named for its inventor, Charles Dodgson (better known as Lewis Carroll). 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.

This algorithm can be described in the following four steps:

One wishes to find

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 ${\displaystyle {\begin{bmatrix}-2&-1\\-1&2\end{bmatrix}}}$, so after dividing we get ${\displaystyle {\begin{bmatrix}8&-2\\-4&6\end{bmatrix}}}$. The process must be repeated to arrive at a 1 × 1 matrix. ${\displaystyle {\begin{bmatrix}{\begin{vmatrix}8&-2\\-4&6\end{vmatrix}}\end{bmatrix}}={\begin{bmatrix}40\end{bmatrix}}.}$ Dividing by the interior of the 3 × 3 matrix, which is just −5, gives ${\displaystyle {\begin{bmatrix}-8\end{bmatrix}}}$ and −8 is indeed the determinant of the original matrix.

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