Determinant - Calculation

Calculation

Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications like checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques. Nonetheless, explicitly calculating determinants is required in some situations, and different methods are available to do so.

Naive methods of implementing an algorithm to compute the determinant include using Leibniz' formula or Laplace's formula. Both these approaches are extremely inefficient for large matrices, though, since the number of required operations grows very quickly: it is of order n! (n factorial) for an n×n matrix M. For example, Leibniz' formula requires to calculate n! products. Therefore, more involved techniques have been developed for calculating determinants.