The Leibniz formula for determinants, named after the German mathematician and philosopher Gottfried Wilhelm Leibniz, is a method for computing the determinant of a square matrix.
A determinant is a scalar value that is uniquely associated with a square matrix, giving important information about its properties and behavior. The Leibniz formula provides a systematic way of calculating this determinant.
In its essence, the formula states that the determinant of an n × n matrix A is the sum of all possible products formed by choosing one entry from each row and column of the matrix, multiplied by the sign (-1) raised to the power of the number of inversions in the permutation that defines the chosen entries. An inversion occurs when a subsequent entry in a permutation is smaller than a preceding entry.
Mathematically, if a matrix A has entries aᵢⱼ, where i and j represent the row and column indices respectively, and the determinant is denoted as |A|, then the Leibniz formula can be written as:
|A| = Σ(± a₁₁⋅a₂ⱼ₂⋅a₃ⱼ₃⋯⋅aₙⱼₙ)
where Σ represents the sum over all possible permutations of the indices j₁, j₂, ..., jₙ, and the sign ± is determined by the number of inversions in the permutation.
The Leibniz formula offers a fundamental method for evaluating determinants, providing a crucial tool for various fields of mathematics and applications, including linear algebra, differential equations, and quantum mechanics, among others.
