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, the Laplace expansion along the ith row is the equality
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.
Similarly, the Laplace expansion along the jth column is the equality
(Each identity implies the other, since the determinants of a matrix and its transpose are the same.)
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.
For clarity we also label the entries of
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.
Using Cauchy's two-line notation, the explicit relation between
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.
is (Notice applying A before B is equivalent to applying inverse of A to the upper row of B in two-line notation) where
Similarly, the result holds if the index of the outer summation was replaced with
[1] Laplace's cofactor expansion can be generalised as follows.
Consider the matrix The determinant of this matrix can be computed by using the Laplace's cofactor expansion along the first two rows as follows.
Firstly note that there are 6 sets of two distinct numbers in {1, 2, 3, 4}, namely let
By defining the complementary cofactors to be and the sign of their permutation to be The determinant of A can be written out as where
In our explicit example this gives us As above, 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.
rows and columns with indices in
The same thing holds for any fixed k columns.
The Laplace expansion is computationally inefficient for high-dimension matrices, with a time complexity in big O notation of O(n!).
Alternatively, using a decomposition into triangular matrices as in the LU decomposition can yield determinants with a time complexity of O(n3).
[2] The following Python code implements the Laplace expansion: