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:
“In democratic ages men rarely sacrifice themselves for another, but they show a general compassion for all the human race. One never sees them inflict pointless suffering, and they are glad to relieve the sorrows of others when they can do so without much trouble to themselves. They are not disinterested, but they are gentle.”
—Alexis de Tocqueville (1805–1859)
“The method of scientific investigation is nothing but the expression of the necessary mode of working of the human mind. It is simply the mode in which all phenomena are reasoned about, rendered precise and exact.”
—Thomas Henry Huxley (1825–95)