Laplace Expansion

linear-algebra

Definition

Laplace Expansion

Laplace expansion is a method for computing the determinant of a square matrix by expanding any row or column. The theorem expresses the determinant of a matrix as a sum of products of its entries and their corresponding cofactors.

Let be a square matrix. The determinant of can be computed by expanding the -th row as follows:

or alternatively, by expanding aloing the -th column:

where is the cofactor and is the minor of .

It was named after Pierre-Simon Laplace.

Example

Consider a matrix :

To compute the determinant of , the row is chosen.

Now, the determinant of can be computed:

Thus, the determinant of is .