Laplace expansion

Laplace expansion expresses a square matrix’s determinant as a signed sum of entries in one row or column multiplied by their corresponding minors. It is also called cofactor expansion.

Cofactor: A matrix entry’s cofactor is its minor multiplied by a sign determined by its row and column. Each term in the expansion is an entry multiplied by its cofactor.
Square matrix: A square matrix has the same number of rows and columns. Laplace expansion applies to square matrices because only they have determinants.
Sparse matrix: A sparse matrix has relatively few nonzero entries compared with its total number of entries. Expanding along a sparse row or column eliminates zero terms.
Gaussian elimination: Gaussian elimination transforms a matrix by row operations to solve linear systems or simplify calculations. It computes determinants in cubic time, unlike the costly repeated expansion of dense matrices.
Minor (linear algebra): A matrix minor is the determinant of a submatrix formed by deleting selected rows and columns. The expansion uses minors obtained by deleting the chosen entry’s row and column.
Submatrix: A submatrix is formed by retaining selected rows and columns of a matrix. Deleting one row and column produces the smaller matrices whose determinants are minors.
Cramer's rule: Cramer's rule solves a square linear system with a nonzero determinant using ratios of determinants. Laplace expansion can evaluate the determinants appearing in the rule.
LU decomposition: LU decomposition factors a matrix into a lower triangular matrix and an upper triangular matrix. The factorization yields determinants efficiently without recursively expanding minors.
Leibniz formula: The Leibniz formula expresses a determinant as a signed sum over permutations of products of matrix entries. Expanding each minor shows how this compact recursion matches the permutation formula.
Alternating multilinear form: An alternating multilinear form is linear in each argument and vanishes when two arguments are equal. These properties underlie determinant formulas and their cofactor signs.

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