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Concept
Laplace expansion
Summary
In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an n × n matrix B as a weighted sum of minors, which are the determinants of some (n − 1) × (n − 1) submatrices of B. Specifically, for every i,
where is the entry of the ith row and jth column of B, and is the determinant of the submatrix obtained by removing the ith row and the jth column of B.
The term is called the cofactor of in B.
The Laplace expansion is often useful in proofs, as in, for example, allowing recursion on the size of matrices. It is also of didactic interest for its simplicity and as one of several ways to view and compute the determinant. For large matrices, it quickly becomes inefficient to compute when compared to Gaussian elimination.
Consider the matrix
The determinant of this matrix can be computed by using the Laplace expansion along any one of its rows or columns. For instance, an expansion along the first row yields:
Laplace expansion along the second column yields the same result:
It is easy to verify that the result is correct: the matrix is singular because the sum of its first and third column is twice the second column, and hence its determinant is zero.
Suppose is an n × n matrix and For clarity we also label the entries of that compose its minor matrix as
for
Consider the terms in the expansion of that have as a factor. Each has the form
for some permutation τ ∈ Sn with , and a unique and evidently related permutation which selects the same minor entries as τ. Similarly each choice of σ determines a corresponding τ i.e. the correspondence is a bijection between and
Using Cauchy's two-line notation, the explicit relation between and can be written as
where is a temporary shorthand notation for a cycle .
This operation decrements all indices larger than j so that every index fits in the set {1,2,...,n-1}
The permutation τ can be derived from σ as follows.
Define by for and .
Official source
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