Laplace expansion

Laplace expansion along the second column yields the same result:

However, Laplace expansion is efficient for small matrices only.

Sarrus' rule can also be derived by looking at the Laplace expansion of a 3x3 matrix.

The Laplace expansion is in fact the expansion of the inverse distance between two points.

Compute determinants of matrices up to order 6 using Laplace expansion you choose.

In general the Slater determinant is evaluated by the Laplace expansion.

For proof, write this homogeneous polynomial as determinants and use Laplace expansion (in reverse).

See also Laplace expansion of determinant.

The determinant of this matrix can be computed by using the Laplace expansion along any one of its rows or columns.

Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:

Finally, the Laplace expansion expresses the determinant in terms of minors, i.e., determinants of smaller matrices.

The and functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions.

The Laplace expansion is of theoretical interest as one of several ways to view the determinant, as well as of practical use in determinant computation.

The Laplace expansion is computationally inefficient for high dimension because for NxN matrices, the computational effort scales with N!

In physics, the Laplace expansion of a 1/r - type potential is applied to expand Newton's gravitational potential or Coulomb's electrostatic potential.

Therefore, the Laplace expansion is not suitable for large N. Using a decomposition into triangular matrices as in the LU decomposition, one can determine determinants with effort N/3.

This can be deduced from some of the properties below, but it follows most easily directly from the Leibniz formula (or from the Laplace expansion), in which the identity permutation is the only one that gives a non-zero contribution.

When linear system solution is introduced via the wedge product, Cramer's rule follows as a side effect, and there is no need to lead up to the end results with definitions of minors, matrices, matrix invertibility, adjoints, cofactors, Laplace expansions, theorems on determinant multiplication and row column exchanges, and so forth.