Leibniz formula for determinants

The determinant of an n × n matrix is the sum over all permutations of products choosing one entry from each row and column, weighted by permutation sign.

Permutation: A bijection from a set to itself, representing a rearrangement of its elements. Each term chooses columns through one permutation of the matrix’s rows.
Square matrix: A matrix with the same number of rows and columns. The formula applies to square arrays of entries.
Cofactor expansion: A method for computing a determinant by expanding along one row or column using signed minors. It can be derived by grouping the permutation sum according to a chosen row or column.
Gaussian elimination: A sequence of elementary row operations that reduces a matrix to echelon form. It computes determinants efficiently without enumerating every permutation.
Permutation sign: The value +1 or −1 assigned to a permutation according to whether it is even or odd. It determines whether each product contributes positively or negatively.
Factorial: For a nonnegative integer n, the product of all positive integers up to n, with 0! = 1. An n × n formula has n! permutation terms.
Cramer’s rule: A formula expressing solutions of a nonsingular linear system as ratios of determinants. The determinant values in its numerator and denominator can be defined by this sum.
Leibniz formula for Pfaffians: A signed sum over pairings that defines the Pfaffian of an even-dimensional skew-symmetric matrix. It resembles the determinant formula but sums over pairings rather than all permutations.
Leibniz expansion: An expansion of a determinant as a sum of signed products, one for each permutation. This is the explicit sum that defines the formula.
Even permutation: A permutation expressible as an even number of transpositions. Even permutations supply the positive terms in the sum.

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Seki TakakazuCompared with: Seki and Gottfried Wilhelm Leibniz developed determinant formulations independently in the seventeenth century.