The General Method
This algorithm can be described in the following 4 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 . 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.
- 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.
- Let A = B, and B = C. Repeat step 3 as necessary until the 1 × 1 matrix is found; its only entry is the determinant.
Read more about this topic: Dodgson Condensation
Famous quotes containing the words general and/or method:
“The world can doubtless never be well known by theory: practice is absolutely necessary; but surely it is of great use to a young man, before he sets out for that country, full of mazes, windings, and turnings, to have at least a general map of it, made by some experienced traveller.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“No method nor discipline can supersede the necessity of being forever on the alert. What is a course of history or philosophy, or poetry, no matter how well selected, or the best society, or the most admirable routine of life, compared with the discipline of looking always at what is to be seen? Will you be a reader, a student merely, or a seer? Read your fate, see what is before you, and walk on into futurity.”
—Henry David Thoreau (1817–1862)